System of First Order Differential Equations


 Marlene Sullivan
 2 years ago
 Views:
Transcription
1 CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions simultaneously. Those unknown functions are related by a set of equations that involving the unknown functions and their first derivatives. For example, in Chapter Two, we studied the epidemic of contagious diseases. Now if S(t) denotes number of people that is susceptible to the disease but not infected yet. I(t) denotes number of people actually infected. R(t) denotes the number of people have recovered. If we assume The fraction of the susceptible who becomes infected per unit time is proportional to the number infected, b is the proportional number. A fixed fraction rs of the infected population recovers per unit time, r. A fixed fraction of the recovers g become susceptible and infected, g. proportional function. The system of differential equations model this phenomena are S = bis + gr I = bis ri R = ri gr The numbers of unknown function in a system of differential equations can be arbitrarily large, but we will concentrate ourselves on to 3 unknown functions.. Principle of superposition Let a ij (t), b j (t) i =,,, n and j =,,, n be known function, and x i t, i =,,, n be unknown functions, the linear first
2 . SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS order system of differential equation for x i (t) is the following, x (t) = a (t)x (t) + a (t)x (t) + + a n (t)x n (t) + b (t) x (t) = a (t)x (t) + a (t)x (t) + + a n (t)x n (t) + b (t) x 3(t) = a 3 (t)x (t) + a 3 (t)x (t) + + a 3n (t)x n (t) + b 3 (t). x n(t) = a n (t)x (t) + a n (t)x (t) + + a nn (t)x n (t) + f (t) Let x(t) be the column vector of unknown functions x i t, i =,,, n, A(t) = (a ij (t), and b(t) be the column vector of known functions b i t, i =,,, n, we can write the first order system of equations as () x (t) = A(t)x(t) + b(t) When n =, the linear first order system of equations for two unknown functions in matrix form is, x (t) a (t) a x = (t) x (t) b (t) + (t) a (t) a (t) x (t) b (t) When n = 3, the linear first order system of equations for three unknown functions in matrix form is, x (t) x (t) = a (t) a (t) a 3 a (t) a (t) a 3 x (t) x (t) + b (t) b (t) x 3(t) a 3 (t) a 3 (t) a 33 x 3 t b 3 (t) A solution of equation () on the open interval I is a column vector function x(t) whose derivative (as a vectorvalues function) equals A(t)x(t) + b(t). The following theorem gives existence and uniqueness of solutions, Theorem.. If the vectorvalued functions A(t) and b(t) are continuous over an open interval I contains t, then the initial value problem { x (t) = A(t)x(t) + b(t) x(t ) = x has an unique vectorvalues solution x(t) that is defined on entire interval I for any given initial value x. When b(t), the linear first order system of equations becomes x (t) = A(t)x(t), which is called a homogeneous equation. As in the case of one equation, we want to find out the general solutions for the linear first order system of equations. To this end, we first have the following results for the homogeneous equation,
3 . PRINCIPLE OF SUPERPOSITION 3 Theorem.. Principle of Superposition Let x (t), bx (t),, x n (t) be n solutions of the homogeneous linear equation x (t) = A(t)x(t) on the open interval I. If c, c,, c n are n constants, then the linear combination is also a solution on I. c x (t) + c x (t) + c 3 x 3 (t) + + c n x n (t) Example.. Let x (t) = x(t) e t, x (t) = and x (t) = e t are two solutions, as (e bx (t) = t ) e t e t = = and [ bx (t) = [ (e t ) = [ e t = [ e t By the Principle of Superposition, for any two constants c and c e t c e x(t) = c x (t) + c x (t) = c + c e t = t c e t is also solution. We shall see that it is actually the general solution. The next theorem gives the general solution of linear system of equations, Theorem.3.  Let x (t), x (t),, bx n (t) be n linearly independent (as vectors) solution of the homogeneous system x (t) = A(t)x(t), then for any solution x c (t) there exists n constants c, c,, c n such that x c (t) = c x (t) + c x (t) + + c n x n (t). We call x c (t) the general solution of the homogeneous system.
4 4. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS If x p (t) is a particular solution of the nonhomogeneous system, x(t) = B(t)x(t) + b(t), and x c (t) is the general solution to the associate homogeneous system, x(t) = B(t)x(t) then x(t) = x c (t) + x p (t) is the general solution. Example.. Let [ 4 3 x 4t (t) = x(t) + + 5t 6 7 6t x(t) + 7t + 3e t e 5t, x (t) = e t and x (t) = 3e 5t are two linearly independent t solutions. and x p (t) = is a particular solution. By Theorem t.3, 3c e x(t) = c x (t) + c x (t) + x p (t) = t + c e 5t + t () c e t + 3c e 5t + t is the general solution. Now suppose we want to [ find a particular solution that satisfies the initial condition x() =, then let t = in (), we have x() = [ 3c + c = c + 3c which can be written in matrix form, [ 3 c = 3 c c Solve this equation, we get = c [ 3e solution is x(t) = t e 5t + t e t 3e 5t. + t, [,. So the particular From the above example, we can summarize the general steps in find a solution to initial value problem, { x (t) = A(t)x(t) + b(t) x(t ) = x
5 . HOMOGENEOUS SYSTEM 5 Step One: Find the general solution x c = c x (t) + c x (t) + + c n x n (t), where x (t), x (t),, x n (t) are a set of linearly independent solutions, to the associate homogeneous system, x (t) = A(t)x(t). Step Two: Find a particular solution x p (t)to the nonhomogeneous system, x (t) = A(t)x(t) + b(t). Step Three: Set x(t) = x c (t) + x p (t) and use the equation x(t ) = x, to determine c, c,, c n.. Homogeneous System We will use a powerful method called eigenvalue method to solve the homogeneous system x (t) = Ax(t) where A is a matrix with constant entry. We will present this method for A is either a or 3 3 cases. The method can be used for A is an n n matrix. The idea is to find solutions of form (3) x(t) = ve λt, a straight line that passing origin in the direction v. Now taking derivative on x(t), we have (4) x (t) = λve λt put (3) and (.) into the homogeneous equation, we get So x (t) = λve λt = Ave λt Av = λv, which indicates that λ must be an eigenvalue of A and v is an associate eigenvector... A is a matrix. Suppose a a A = a a Then the characteristic polynomial p(λ) of A is p(λ) = A λi = (a λ) (a λ) a a = λ (a +a )+(a a a a. So p(λ) is a quadratic polynomial of λ. From Algebra, we know that p(λ) = has either distinct real solutions, or a double solution, or conjugate complex solutions. The following theorem summarize the solution to the homogeneous system,
6 6. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS Theorem.. Let p(λ) be the characteristic polynomial of A, for x (t) = Ax(t), Case : p(λ) = has two [ distinct real solutions [ λ and λ. v v Suppose v = and v v = are associate eigenvector (i.e, Av = λ v and Av = λ v ) Then the general v solution is And x c (t) = c v e λ t + c v e λ t v e Φ(t) = λ t v e λ t v e λ t v e λ t is called the fundamental matrix(a fundamental matrix is a square matrix whose columns are linearly independent solutions of the homogeneous system). Case : p(λ) = has a double solutions λ. In this case p(λ) = (λ λ ) and λ is a zero of p(λ) with multiplicity. () λ has [ two linearly independent eigenvectors: v v Suppose v = and v v = are associate linearly v independent eigenvectors. Then the general solution is And x c (t) = (c v + c v )e λ t Φ(t) = e λ t v v v v () λ has [ only one associate eigenvector: v Suppose v = is the only associated eigenvector and v v v = is a solution of v (λ I A)v = v. Then the general solution is, And x c (t) = (c v + c (tv + v )e λ t [ Φ(t) = e λ t v (v t + v ) v (v t + v ) is the fundamental solution matrix.
7 . HOMOGENEOUS SYSTEM 7 Case 3: p(λ) = has two [ conjugate complex solutions a+bi and a bi. v + iv Suppose v = is the associate complex eigenvector[ with respect to a [ + bi, then the general solution is, v + iv v v v = and v v = v x c (t) = [c (v cos(bt) v sin(bt))c (v cos(bt) + v sin(bt))e at. And [ Φ(t) = e at v cos(bt) v sin(bt) v cos(bt) + v sin(bt) v cos(bt) v sin(bt) v cos(bt) + v sin(bt) is the fundamental matrix. From Theorem., let Φ(t) be the fundamental [ matrix, the general c solution is given by x c (t) = Φ(t)c, with c = and the solution c that satisfies a given initial condition x(t ) = x is given by x(t) = Φ(t)Φ(t ) t x Example.. [ Two distinct eigenvalues case Find the general 3 solution to x (t) = x(t) 5 Using Mathcad, functions eigenvals() and eigen Solution vecs() In Mathcad, eigenvecs(m) Returns a matrix containing the eigenvectors. The nth column of the matrix returned is an eigenvector corresponding to the nth eigenvalue returned by eigenvals. we find,λ = [ and λ = 3 6 with associated eigenvectors v = and v [ = respectively. So the fundamental matrix is [ ( 7 6)e ( 3 Φ(t) = + 6)t ( 7 + 6)e ( 3 6)t e ( 3 + 6)t e ( 3 6)t c and the general solution is, for c =, c x c (t) = Φ(t)c
8 8. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS Example.. One double eigenvalues with two linearly [ independent eigenvectors Find the general solution to x (t) = x(t). Solution [ The eigenvalue is λ = and associated [ eigenvectors c e are and, so the general solution is x c = t c e t Example.3. One double eigenvalues with only one eigenvector Find the solution to x (t) = x(t) and x() = Solution Using Mathcad, functions eigenvals() and [ eigenvecs() we can find a double eigenvalue λ = 5 and eigenvector Notice, the symbolic operator (bring up by either [Shift[Ctrl[. or [Ctrl[.) will not work with eigenvecs() this time, but since multiply an eigenvector by a nonzero constant still get an eigenvector, we can 3 choose v =. 3 To [ find w that [ satisfies (A λ I)w = v λ we will solve (A w λ I) =. That is, w 3 3 w 3 = 3 3 w 3 One solution is w = and w = So the fundamental matrix is 3 3t + Φ(t) = e 5t 3 3t c and the general solution is, c =, c x c (t) = Φ(t)c 3 3() + Now, Φ() = e 5() = 3 3() [ 3 3 and Φ()  = Hence, the particular solution is x(t) = Φ(t)Φ()  x = e 5t [ 3t + 4 3t 3.
9 . HOMOGENEOUS SYSTEM 9 Example.4. Two conjugate [ complex eigenvalues case Find 3 the general solution to x (t) = x(t) Solution Using Mathcad, functions eigenvals() and eigenvecs() we find two conjugate complex eigenvalues, λ = + i 3 and λ = i [ 3 3 with associated eigenvector v = with respect to i λ. Compare this with the Theorem., we have a =, b = 3, v = 3, v =, v =, and v =. So the fundamental matrix is [ 3 cos(bt) sin(bt) Φ(t) = e t 3 sin(bt) cos(bt) and the general solution is, c = [ c c, [ 3 cos( 3t) sin( 3t) x c (t) = Φ(t)c = e t c 3 sin( 3t) cos( 3t) c [ = e t 3c cos( 3t) c sin( 3t) 3c sin( 3t) c cos( 3t) Suppose we want to find a solution such that x() =, then x(t) = Φ(t)Φ()  x() [ [ 3 cos( 3t) sin( 3t) = e t 3 [ 3 sin( 3t) cos( 3t) [ 3 cos( 3t) sin( 3t) 3 = e t 3 sin( 3t) cos( 3t) = e t [ cos( 3t) + + sin( 3t) sin( 3t) + cos( 3t).. A is a 3 3 matrix. Suppose A = a a a 3 a a a 3 a 3 a 3 a 33 Then the characteristic polynomial p(λ) of A given by p(λ) = A λi,  [
10 . SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS is a cubic polynomial of λ. From Algebra, we know that p(λ) = has either 3 distinct real solutions, or distinct solutions and one is a double solution, or one real solution and conjugate complex solutions, or a triple solution. The following theorem summarize the solution to the homogeneous system, Theorem.. Let p(λ) be the characteristic polynomial of A, for x (t) = Ax(t), Case : p(λ) = has three distinct real solutions λ, λ, and λ 3. Suppose v = v v, v = v v, and v 3 = v 3 v 3 v 3 are associate eigenvector (i.e, Av = λ v, Av = λ v, and Av 3 = λ 3 v 3 ) Then the general solution is v 3 x c (t) = c v e λ t + c v e λ t + c 3 v 3 e λ 3t And the fundamental matrix is Φ(t) = v e λ t v e λ t v e λ t v e λ t v 3 e λ 3t v 3 e λ 3t v 3 e λ t v 3 e λ t v 33 e λ 3t Case : p(λ) = has a double solutions λ. So p(λ) = (λ λ ) (λ λ ), and λ has multiplicity. Let v 3 = v v is the eigenvector associated with λ. v 3 [ λ has [ two linearly independent eigenvectors: v v Suppose v = and v v = are associate linearly v independent eigenvectors. Then the general solution is And x c (t) = (c v + c v )e λ t + c 3 v 3 e λ t Φ(t) = v e λ t v e λ t v e λ t v e λ t v 3 e λ t v 3 e λ t v 3 e λ t v 3 e λ t v 33 e λ t. v 33
11 . HOMOGENEOUS SYSTEM [ λ has one eigenvector: Suppose v = v v is the associated eigenvector with respect to λ and v = v v 3 v is a solution of v 3 (λ I A)v = v. Then the general solution is, x c (t) = (c v + c (tv + v ))e λ t + c 3 v 3 e λ And Φ(t) = v e λ t v e λ t (v t + v )e λ t (v t + v )e λ t v 3 e λ v 3 e λ v 3 e λ t (v 3 t + v 3 )e λ t v 33 e λ is the fundamental solution matrix. Case 3: p(λ) = has two conjugate complex solutions a ± bi and a real solution λ. Suppose v = v + iv v + iv is the associate complex eigenvector with respect to a + bi, then the general solution is, let v 3 + iv 3 v 3 = v 3 v 3, are associated eigenvectors with respect to λ, V 33 x c (t) = [c (v cos(bt) v sin(bt))c (v cos(bt)+v sin(bt))e at +c 3 v 3 e λ. Φ(t) = e at And v cos(bt) v sin(bt) v cos(bt) + v sin(bt) v 3 e λ v cos(bt) v sin(bt) v cos(bt) + v sin(bt) v 3 e λ v 3 cos(bt) v 3 sin(bt) v 3 cos(bt) + v 3 sin(bt) v 33 e λ is the fundamental matrix. Case 4: p(λ) = has solution λ with multiplicity 3. In this case, p(λ) = (λ λ ) 3. [ λ has three linearly independent eigenvectors. Let v = v v v 3, v = v v V 3, and v 3 = v 3 v 3 V 33 be the three linearly independent eigenvectors. Then the general
12 . SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS solution is x c (t) = (c v + c v + c 3 v 3 )e λt and fundamental v v v 3 v v V 3 matrix is Φ(t) = e λ t v 3 v 3 V 33 [ λ has two linearly independent eigenvectors. Suppose v = v v, v = v v are the linearly independent eigenvectors. Let v 3 = v v 3 V 3 3 v 3, then only one of the two equations, (A λ I)v 3 = v or (A λ I)v 3 = v can has a solution that is linearly independent with v, v. Suppose (A λ I)v 3 = v generates such a solution. Then the general solution is x c (t) = [c v + c v + c 3 (tv + v 3 )e λ t and fundamental matrix is Φ(t) = e λ t v v tv + v 3 v v tv + V 3 v 3 v 3 tv 3 + V 33 [3 λ has only one eigenvector. Let v = v v be the linearly independent eigenvectors. Let v = satisfies v 3 v v V 3 and v 3 = V 33 v 3 v 3 V 33 (A λ I)v = v and (A λ I)v 3 = v. be two vectors that Then the general solution is x c (t) = [c v + c (tv + v ) + c 3 (t v + tv + v 3 )e λ t and fundamental matrix is Φ(t) = e λ t v tv + v t v + tv + v 3 v tv + v t v + tv + V 3 v 3 tv 3 + v 3 t v 3 + tv 3 + V 33 Remark.. Suppose A is an n n matrix, for the homogeneous system x (t) = Ax(t), three general case would happen Case : A has n distinct eigenvalues λ i, i =,,, n with linearly independent eigenvectors v i, i =,,, n then the general solution will be x c (t) = c v e λ + c v e λ + + c n v n e λn
13 . HOMOGENEOUS SYSTEM 3 Case: A has m < n distinct eigenvalues, in this case some eigenvalues would have multiplicity greater than. Suppose λ r has multiplicity r. Depending on how many linearly independent eigenvectors are associated with λ r the situation could be very complex. Let p be the number of linearly eigenvectors associated with λ r, then d = r p is called the deficit of λ r. The simply cases are either d = or d = r. When < d < r the situation could be very complex. Suppose d = r and v is the only eigenvector associate with λ r, then one will have to solve r equations (A λ r ) i v i+ = v i, i =,,, r. And the general solution would contains terms like [c v + c (v t + v ) + c 3 (v t + v t + v 3 ) + + c r (v r + v t r + + v r )e λr. Case 3: A complex root a+bi with associated eigenvector v a +iv b, then the general solution contains term, [c (v a cos(bt) v b sin(bt))+ c (v a sin(bt) + v b cos(bt))e at. Remark.. Suppose x (t), x (t), x 3 (t),, x n (t) are n linearly independent solution for n n homogeneous system, x (t) = Ax(t), the fundamental matrix Φ(t) is a matrix whose columns are x i (t), i =,,, n. Example.5. (Two distinct eigenvalues) Find the general solution to x = 3x + 4x x 3 x = x + x 4x 3 x 3 = x + x Solution Let x(t) = x (t) x (t) x 3 (t) and The equations can be written in matrix form x (t) = Ax(t). Using Mathcad, functions eigenvals() and eigenvecs() we find,λ = and λ = with associated eigenvectors v = 4 and v = respectively. Since λ has multiplicity as appeared twice in the result of eigenvals() function, we need to solve the equation (A λ I)v 3 = v.
14 4. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS To use Mathcad, () you first compute (A λ I)v 3 using the following sequences of key stroke, [* type ([Ctrl[M, set the rows and columns in the matrix definition popup menu, input the data for A, [* type [Ctrl[M set the row and column number and input data for λ I, [* type )[Ctrl[M, now set as column number, enter a, b, c in the place holders, [* type [Ctrl[. to compute symbolically and you get. () Using the Given Find block to find a solution. Type Given in a blank space, type a+bc[ctrl= and a4c[ctrl= in two rows, then type key word Find following by typing (a,b)[ctrl[. you will get the solution in terms of c. Set c =, we get v 3 = 4. So the fundamental matrix is Φ(t) = and the general solution is, 4et e t (t + 4)e t e t e t e t e t (t + )e t x c (t) = c v e t + c v e t + c 3 (tv + v 3 )e t Example.6. (One eigenvalue with deficit ) Find the solution to x (t) = 3 x(t) and x() = Solution Using Mathcad, functions eigenvals() (Notice the eigenvecs() will not find a good result in this case due to the rounding error.) we find, λ = is the only eigenvalue. To find the associate eigenvectors we compute (Using (A λ I)v = ) (A λ I)v = v v 4 v 3 v + v 3 v + v 3 = 4v v 3
15 . HOMOGENEOUS SYSTEM 5 We have only v + v 3 = for three variables v, v, v 3, this indicates that v can be any value, and set v = find v 3 =, So v = and v = are two eigenvectors. To find the generalize eigenvector associated with λ we will have to solve two equations (A λ I) w w =, w 3 and From (.), (A λ I) 4 w w w 3 = [ w w =, we get two inconsistent equations w + w 3 = and w + w 3 =. So now solution can be found in this case. From (.), 4 [ w w = we get one equation w + w 3 = choose w 3 = we get w =, since w can be anything, we set w =. So v 3 = and we can verify that v, v, and v 3 are linearly independent. So the fundamental matrix is Φ(t) = e t t t + t + and the general solution is,,, x c (t) = [c v + c v + c 3 (tv + v 3 )e t
16 6. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS Now, Φ() = and Φ() = Hence, the particular solution is x(t) = [v + 3v e t 3. Example.7. (One eigenvalue with deficit ) Find the general solution to x (t) = 4 x(t). 3 Solution Using Mathcad, functions eigenvals() (Notice the eigenvecs() will not find a good result in this case due to the rounding error.) we find, λ = 3 is the only eigenvalue. To find the associate eigenvectors we compute (Using (A λ I)v = ) (A 3I)v = v v v 3 v + v + v 3 v + v + v 3 = v v We have only one eigenvector v =. To find the generalize eigenvector associated with λ we will have to solve two equations and From (.), (A 3I)v = v, (A 3I)v 3 = v, a b c =, we have two equations { b a = a + b + c = Choosing a =, we get b =, c =. Hence v =
17 . HOMOGENEOUS SYSTEM 7 From (.), a b c =, we have two equations { b a = a + b + c = Choosing a =, we get b =, c =. So v 3 = that v, v, and v 3 are linearly independent. So the fundamental matrix is Φ(t) = e t + t t + t t t t t + and the general solution is, and we can verify x c (t) = [c v + c (tv + v ) + c 3 (t v + tv + v 3 )e 3t Example.8. (Two conjugate complex eigenvalues case) Find the general solution to x (t) = 3 3 x(t) Solution Using Mathcad, functions eigenvals() and eigenvecs() we find two conjugate complex eigenvalues and one real eigenvalue, λ =, λ = + i 9, and λ3 = i 9 with associated eigenvector v = and v = + i 9 + i 9 = + 9 i 9 with respect to λ 3. Compare this with the Theorem.3,
18 8. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS we have a =, b = 9, v = 9 9 (imaginary part of v). The general solution is, x c (t) = c v e t +c (v cos( (real part of v), and v 3 = 9) v3 sin( 9))e t +c 3 (v sin( 9)+v3 cos( 9))e t 3. Nonhomogeneous System of Equations To find solutions to the initial value problem of nonhomogeneous equations x (t) = Ax(t) + b(t), x(t ) = x we follow the steps below, () Find the general solution x c (t) = Φ(t)c to homogeneous equation x (t) = Ax(t), where Φ(t) is the fundamental matrix. () Find a particular solution x p to x (t) = Ax(t) (3) The general solution to the nonhomogeneous equation x (t) = Ax(t) is x(t) = x c (t) + x p (t). Using x(t ) = x to determine the coefficient vector c. The following theorem gives one way to find a particular solution based on the fundamental matrix, Theorem 3.. Let Φ(t) be a fundamental matrix of x (t) = Ax(t), a particular solution to x (t) = Ax(t) + b(t) is given by x p (t) = Φ(t) Φ(t)  b(t) dt. Example 3.. Find the general solution to Solution Let x(t) = b(t) = t t x = 3x + 4x x 3 + t x = x + x 4x 3 x 3 = x + x t x (t) x (t) x 3 (t), A = 3 4 4, and. The equations can be written in matrix form x (t) =
19 3. NONHOMOGENEOUS SYSTEM OF EQUATIONS 9 Ax(t)+b(t). From Example.5, we know that the fundamental matrix to x(t) = Ax(t) is Φ(t) = 4et e t (t + 4)e t e t e t e t e t (t + )e t To find a particular solution, we first compute Φ  (t)b(t) = then we compute Φ(t) Φ  5 (t)b(t) dt = Φ(t) t e t dt ( 5 t3 + 5 t )e t dt 5 t e t dt 5 = t + t t 3t t + 4t And so the general solution is, x(t) = c 4 e t + c + c 3 (t + 4 e t ) e t + 5 t + t t 3 5 t t t The following is a screen shot that shows how to carry out the computation in Mathcad, 5 t e t ( 5 t3 + 5 t )e t 5 t e t To use Mathcad, () Define fundamental matrix A(t) and b(t) in the same line (not as shown in graph), and compute in the next line A b(t) () type A(t)*[Ctrl[M choose column as, at each place holder, type [Ctrl[I to get the indefinite integral, (3) and put the corresponding entry of A b(t) in the integrant position. (4) press [Shift[Ctrl[. type key work simplify
20 . SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS Theorem 3.. If Φ(t) is the fundamental matrix for x (t) = Ax(t),, and x p (t) = Φ  (t)b(t) dt, then x(t) = Φ(t)Φ  (t )(x x p (t )) + x p (t) is the solution to the nonhomogeneous initial value problem, x (t) = Ax(t) + b(t), x(t ) = x Example 3.. Find the solution to x (t) = e t e t and x() = [ x(t) + Solution From Example.3 is the fundamental matrix is 3 3t + Φ(t) = e 5t 3 3t and Φ()  = e Now b(t) = t e t, using the formula x p (t) = Φ(t) Φ  (t)b(t) dt and Mathcad, we have x p (t) = 3 et Therefore, Φ()  (x() x p ()) = ( [ and the solution is x(t) = Φ(t)Φ()  (x() x p ()) + x p (t) = e 5t [ [ 3 ) = [ t 6 4t + 3 e 3t 4. Higher order differential equations One can transform equations that involving higher order derivatives of unknown functions to system of first order equations. For example, suppose x(t) is an unknown scalar function that satisfies mx (t) + cx (t) + kx(t) = f(t) an equation can be used to model a spring system with external force f(t) or an RCL electronic circuit with an energy source f(t).
21 4. HIGHER ORDER DIFFERENTIAL EQUATIONS Now if we set x (t) = x(t) and x (t) = x (t) we then get an system of first order equations (5) x (t) = x (t) (6) x (t) = c m x (t) k m x (t) + f(t) m In general, if we have an differential equation that involving nth order derivative x (n) (t) of unknown function x(t), x (n) = a x(t) + a x (t) + + a n x (n ) + f(t), we can transform it into an system of first order equations of n unknown functions x (t) = x(t), x (t) = x (t), x 3 (t) = x () (t),, x n (t) = x (n ) (t), and using the eigenvalue method for system of differential equation to solve the higher order equation. Example 4.. Transform the differential equation x (3) + 3x () 7x (t) 9x = sin(t) into system of first order equations. Solution Here the highest order of derivative is third derivative x (3) of x(t). So we transfer it into system of 3 equations. Let x (t) = x(t), x (t) = x (t), x 3 (t) = x (t), we have (7) (8) x (t) = x (t) x (t) = x 3 (t) (9) x 3(t) = 3x 3 (t) + 7x (t) + 9x (t) sin(t) Let x(t) = x (t) x (t), A =, and b(t) = x 3 (t) f(t) we can write the system of equation in matrix form x (t) = Ax(t) + b(t). Example 4.. Find the general solution for the 3rd order differential equation x (3) + 3x () 7x (t) 9x = sin(t). Solution From previous example, Example 4., Let x(t) = x (t) x (t), A =, and b(t) = we can write x 3 (t) f(t) the system of equation in matrix form x (t) = Ax(t) + b(t). Using Mathcad we find the eigenvalues are λ =, λ = +, λ 3 = with associate eigenvectors, v =, v = +,
22 . SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS and v 3 = + respectively (after multiply the results of Mathcad by some constants). So the fundamental matrix is Φ(t) = e t e ( + )t e (+ )t e t ( + )e ( + )t ( + )e (+ )t e t ( )e ( + )t ( + )e (+ )t From Φ(t) we find a particular solution 3 x p (t) = Φ(t) Φ  5 (t)b(t) dt = cos(t) 5 sin(t) cos(t) sin(t) cos(t) 8 sin(t) 5 39 Hence the general solution to the system is x (t) x (t) = x 3 (t) c e t + c e ( + )t + c 3 e (+ )t cos(t) 5 39 sin(t) c e t + c ( + e ( + )t c 3 ( + )e (+ )t cos(t) 56 sin(t) c e t + c ( )e ( + )t + c 3 ( + )e (+ )t 3 5 cos(t) 8 39 sin(t) and x (t) = c e t + c e ( + )t + c 3 e (+ )t + 3 cos(t) 5 sin(t) is 5 39 the general solution to the third order ordinary differential equation x (3) + 3x () 7x (t) 9x = sin(t). Example 4.3. Find the solution to the initial value problem x x + 9x = te t, x() =, x () =. Solution Since the given equation is of second order, we will have two unknowns x (t) = x(t), x (t) = x (t) to transform the equation into a system of first order equations, x (t) = x x (t) = x 9x + te t, and the initial conditions [ are x () = [ x() = x () = x () = [. x (t) Now let x(t) =, A =, and b(t) = x (t) 9 te t. We have the matrix version of this equation, x (t) = Ax(t) + b(t)
23 4. HIGHER ORDER DIFFERENTIAL EQUATIONS 3 Using Mathcad, we find[ the eigenvalues [ λ =, λ = 9, and associate eigenvectors v =, v =. And fundamental 9 e t e matrix Φ(t) = 9t e t 9e 9t. From Φ(t) we find a particular solution x p (t) = Φ(t) Φ  (t)b(t) dt = 5 (3t + 8t + )e t 5 (3t + 7t + 9)e t The solution with initial values x = is given by x(t) = Φ(t)Φ (  ()(x b()) + ) b(t) = 6 t ( 6 t e t 7 5 e9t ) e t 43 5 e9t Hence the solution to the initial value ( problem of the) second order differential equation is x(t) = x (t) = 6 t 639 t+ e t e9t. Project At beginning you should enter: Project title, your name, ss#, and due date in the following format Project One: Define and Graph Functions. John Doe SS#  Due: Mon. Nov. 3rd, 3 You should format the text region so that the color of text is different than math expression. You can choose color for text from Format >Style select normal and click modify, then change the settings for font. You can do this for headings etc. () Solutions To System of Equations Finding solution to linear system using Mathcad and study the long time dynamic behavior of the solutions. Find general solution to { x = y y = x
24 4. SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS Plot several solutions with different initial values in [ xtplane, ytplane xyplane. Here you will need to define range variable t =,. 7 and set X := x(t), Y := y(t). The graph in xyplane is called the trajectory. If this models the movement of a satellite, what is its trajectory. Find general solution to { x = 8y y = 8x Plot several solutions with different initial values in [ xtplane, ytplane xyplane. Here you will need to define range variable t =,. 7 and set X := x(t), Y := y(t). The graph in xyplane is called the trajectory. If this models the movement of a satellite, what is its trajectory. Find general solution to { x = x y y = y 3x Plot several solutions with different initial values in [ xtplane, ytplane xyplane. Here you will need to define range variable t =,. 7 and set X := x(t), Y := y(t). The graph in xyplane is called the trajectory. If this models the movement of a system of two species, what is you conclusion about interdependency of these species? Can you find initial value such that x(t) = (distinct) for some t? what about y(t). () Solution of Higher order equation In general mx + cx + kx = f(t) models a object with mass m attached to a spring with constant k and damping force that is proportional to the velocity x, c, k >. Suppose m = and f(t) = Ae at sin(bt), that is the external force is oscillatory (b > ) and diminishing (a > ) Find solutions and graph the solutions.  c = b = Find general solution and graph some particular solutions.  c =, k =, a =, b =, A = 4  c =, k = 3, A =, a =, b =
Brief Introduction to Vectors and Matrices
CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vectorvalued
More information22 Matrix exponent. Equal eigenvalues
22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to
More informationIntroduction to Mathcad
CHAPTER 1 Introduction to Mathcad Mathcad is a product of MathSoft inc. The Mathcad can help us to calculate, graph, and communicate technical ideas. It lets us work with mathematical expressions using
More informationMatrix Methods for Linear Systems of Differential Equations
Matrix Methods for Linear Systems of Differential Equations We now present an application of matrix methods to linear systems of differential equations. We shall follow the development given in Chapter
More informationr (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)
Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationHigher Order Linear Differential Equations with Constant Coefficients
Higher Order Linear Differential Equations with Constant Coefficients Part I. Homogeneous Equations: Characteristic Roots Objectives: Solve nth order homogeneous linear equations where a n,, a 1, a 0
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationA Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of MatrixValued Functions of a Real Variable
More informationPartial Fraction Decomposition for Inverse Laplace Transform
Partial Fraction Decomposition for Inverse Laplace Transform Usually partial fractions method starts with polynomial long division in order to represent a fraction as a sum of a polynomial and an another
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationC 1 x(t) = e ta C = e C n. 2! A2 + t3
Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationHigher Order Equations
Higher Order Equations We briefly consider how what we have done with order two equations generalizes to higher order linear equations. Fortunately, the generalization is very straightforward: 1. Theory.
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More informationEIGENVALUES AND EIGENVECTORS
Chapter 6 EIGENVALUES AND EIGENVECTORS 61 Motivation We motivate the chapter on eigenvalues b discussing the equation ax + hx + b = c, where not all of a, h, b are zero The expression ax + hx + b is called
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationNON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that
NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called
More informationMath Department Student Learning Objectives Updated April, 2014
Math Department Student Learning Objectives Updated April, 2014 Institutional Level Outcomes: Victor Valley College has adopted the following institutional outcomes to define the learning that all students
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationEXAM. Practice Questions for Exam #2. Math 3350, Spring April 3, 2004 ANSWERS
EXAM Practice Questions for Exam #2 Math 3350, Spring 2004 April 3, 2004 ANSWERS i Problem 1. Find the general solution. A. D 3 (D 2)(D 3) 2 y = 0. The characteristic polynomial is λ 3 (λ 2)(λ 3) 2. Thus,
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006
MATH 511 ADVANCED LINEAR ALGEBRA SPRING 26 Sherod Eubanks HOMEWORK 1 1.1 : 3, 5 1.2 : 4 1.3 : 4, 6, 12, 13, 16 1.4 : 1, 5, 8 Section 1.1: The EigenvalueEigenvector Equation Problem 3 Let A M n (R). If
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...
More informationLecture 5 Rational functions and partial fraction expansion
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions polezero plots partial fraction expansion repeated poles nonproper rational functions
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationMath 2280 Section 002 [SPRING 2013] 1
Math 2280 Section 002 [SPRING 2013] 1 Today well learn about a method for solving systems of differential equations, the method of elimination, that is very similar to the elimination methods we learned
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationMath 2280  Assignment 6
Math 2280  Assignment 6 Dylan Zwick Spring 2014 Section 3.81, 3, 5, 8, 13 Section 4.11, 2, 13, 15, 22 Section 4.21, 10, 19, 28 1 Section 3.8  Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue
More informationEigenvalues and eigenvectors of a matrix
Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a nonzero column vector V such that AV = λv then λ is called an eigenvalue of A and V is
More informationStudent name: Earlham College. Fall 2011 December 15, 2011
Student name: Earlham College MATH 320: Differential Equations Final exam  In class part Fall 2011 December 15, 2011 Instructions: This is a regular closedbook test, and is to be taken without the use
More information19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style
Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationMath 1111 Journal Entries Unit I (Sections , )
Math 1111 Journal Entries Unit I (Sections 1.11.2, 1.41.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More information9.3 Advanced Topics in Linear Algebra
548 93 Advanced Topics in Linear Algebra Diagonalization and Jordan s Theorem A system of differential equations x = Ax can be transformed to an uncoupled system y = diag(λ,, λ n y by a change of variables
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationSo far, we have looked at homogeneous equations
Chapter 3.6: equations Nonhomogeneous So far, we have looked at homogeneous equations L[y] = y + p(t)y + q(t)y = 0. Homogeneous means that the right side is zero. Linear homogeneous equations satisfy
More informationNonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve secondorder nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
More information7  Linear Transformations
7  Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationIterative Methods for Computing Eigenvalues and Eigenvectors
The Waterloo Mathematics Review 9 Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca Abstract: We examine some numerical iterative
More informationDynamics. Figure 1: Dynamics used to generate an exemplar of the letter A. To generate
Dynamics Any physical system, such as neurons or muscles, will not respond instantaneously in time but will have a timevarying response termed the dynamics. The dynamics of neurons are an inevitable constraint
More informationModule 3F2: Systems and Control EXAMPLES PAPER 1  STATESPACE MODELS
Cambridge University Engineering Dept. Third year Module 3F2: Systems and Control EXAMPLES PAPER  STATESPACE MODELS. A feedback arrangement for control of the angular position of an inertial load is
More informationLinearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for inclass presentation
More information3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes
Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationwith "a", "b" and "c" representing real numbers, and "a" is not equal to zero.
3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,
More informationSection 25 Quadratic Equations and Inequalities
5 Quadratic Equations and Inequalities 5 a bi 6. (a bi)(c di) 6. c di 63. Show that i k, k a natural number. 6. Show that i k i, k a natural number. 65. Show that i and i are square roots of 3 i. 66.
More informationComplex Eigenvalues. 1 Complex Eigenvalues
Complex Eigenvalues Today we consider how to deal with complex eigenvalues in a linear homogeneous system of first der equations We will also look back briefly at how what we have done with systems recapitulates
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More informationLecture 31: Second order homogeneous equations II
Lecture 31: Second order homogeneous equations II Nathan Pflueger 21 November 2011 1 Introduction This lecture gives a complete description of all the solutions to any differential equation of the form
More information1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
More informationSystems of Linear Equations
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC
More informationChapter 4  Systems of Equations and Inequalities
Math 233  Spring 2009 Chapter 4  Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationPractice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.
Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationPartial Fractions: Undetermined Coefficients
1. Introduction Partial Fractions: Undetermined Coefficients Not every F(s) we encounter is in the Laplace table. Partial fractions is a method for rewriting F(s) in a form suitable for the use of the
More informationAdvanced Higher Mathematics Course Assessment Specification (C747 77)
Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationThe Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations
The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential
More informationAlgebra. Indiana Standards 1 ST 6 WEEKS
Chapter 1 Lessons Indiana Standards  11 Variables and Expressions  12 Order of Operations and Evaluating Expressions  13 Real Numbers and the Number Line  14 Properties of Real Numbers  15 Adding
More informationHW6 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) November 14, 2013. Checklist: Section 7.8: 1c, 2, 7, 10, [16]
HW6 Solutions MATH D Fall 3 Prof: Sun Hui TA: Zezhou Zhang David November 4, 3 Checklist: Section 7.8: c,, 7,, [6] Section 7.9:, 3, 7, 9 Section 7.8 In Problems 7.8. thru 4: a Draw a direction field and
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More information1.1 Solving a Linear Equation ax + b = 0
1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x = 0 x = (ii)
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationFurther Maths Matrix Summary
Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number
More information9. Particular Solutions of Nonhomogeneous second order equations Undetermined Coefficients
September 29, 201 91 9. Particular Solutions of Nonhomogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
55 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 we saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c n n c n n...
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationIntroduction to polynomials
Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os
More informationSection 7.4 Matrix Representations of Linear Operators
Section 7.4 Matrix Representations of Linear Operators Definition. Φ B : V à R n defined as Φ B (c 1 v 1 +c v + + c n v n ) = [c 1 c c n ] T. Property: [u + v] B = [u] B + [v] B and [cu] B = c[u] B for
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More information