Sample Solutions of Assignment 5 for MAT3270B: Section 3.4. In each of problems find the general solution of the given differential equation
|
|
- Valentine O’Neal’
- 7 years ago
- Views:
Transcription
1 Sample Solutions of Assignment 5 for MAT370B: Section 3.4 In each of problems find the general solution of the given differential equation 7. y y + y = y + 9y = y + 9y 4y = y + y + 1.5y = 0 Answer: 7. The characteristic equation is r + = 0 Thus the possible values of r are r 1 = 1 + i and r = 1 i, and the general solution of the equation is y(t) = e t (c 1 cos t + c sin t). 1. The characteristic equation is 4r + 9 = 0 Thus the possible values of r are r 1 = 3i and r = 3i, and the general solution of the equation is y(t) = c 1 cos 3t + c sin 3t. 14. The characteristic equation is 9r + 9r 4 = 0 1
2 Thus the possible values of r are r 1 = 1 and r 3 = 4, and the general 3 solution of the equation is y(t) = c 1 e t 3 + c e 4t The characteristic equation is r + r = 0 Thus the possible values of r are r 1 = 1 + iand r = 1 i, and the general solution of the equation is y(t) = e t (c1 cos t + c sin t). 3. Consider the initial value problem 3u u + u = 0, u(0) =, u (0) = 0. a. Find the solution u(t) of this problem. b. Find the first time at which u(t) = 10. Answer: (a.) The characteristic equation is 3r r + = 0 Thus the possible values of r are r 1 = 1+ 3i and r 6 = 1 3i, and the 6 general solution of the equation is u(t) = e t 3t 3t 6 (c1 cos + c sin 6 6 ). (b.) From u(0) = 0, we get c 1 = From u (0) = 0, we get c = 3
3 3 Therefore, u(t) = e t 6 ( cos 3t 6 3t sin 3 6 ). The first time is = such that u(t) = Show that W (e λt cos µt, e λt sin µt) = µe λt Answer: Let y 1 = e λt cos µt y = e λt sin µt W (e λt cos µt, e λt sin µt) = y y 1 y 1y = e λt cos µt(λ sin µt + µ sin µt) e λt sin µt(λ cos µt µ sin µt) = µe λt 33. If the functions y 1 and y are linearly independent solutions of y + p(t)y + q(t)y = 0, show that between consecutive zeros of y 1 there is one and only one zero of y. Note that this result is illustrated by the solutions y 1 = cos t and y = sin t of the equation y + y = 0. Answer: Assume the two consecutive zeros of y 1 are t 1 and t, and t 1 < t. Then y 1 (t) < 0 or y 1 (t) > 0 for all t 1 < t < t. Case A: We consider the case y 1 (t) > 0 for all t 1 < t < t. In this case, obviously, y 1(t 1 ) > 0 and y 1(t ) < 0.
4 4 W (y 1, y ) = y y 1 y 1y 0 for all t, because y 1 and y are linearly independent solutions of y + p(t)y + q(t)y = 0, then W (y 1, y )(t) > 0 or W (y 1, y )(t) < 0 for all t. If W (y 1, y )(t) > 0 for all t. We get W (y 1, y )(t 1 ) = y 1(t 1 )y (t 1 ) > 0 W (y 1, y )(t ) = y 1(t )y (t ) > 0 Hence y (t 1 ) > 0 and y (t ) < 0, so there is one zero of y. If W (y 1, y )(t) < 0 for all t. We get W (y 1, y )(t 1 ) = y 1(t 1 )y (t 1 ) < 0 W (y 1, y )(t ) = y 1(t )y (t ) < 0 Hence y (t 1 ) < 0 and y (t ) > 0, so there is one zero of y. Case B: We consider the case y 1 (t) < 0 for all t 1 < t < t. In this case, obviously, y 1(t 1 ) < 0 and y 1(t ) > 0. W (y 1, y ) = y y 1 y 1y 0 for all t, because y 1 and y are linearly independent solutions of y + p(t)y + q(t)y = 0, then W (y 1, y )(t) > 0 or W (y 1, y )(t) < 0 for all t. If W (y 1, y )(t) > 0 for all t. We get W (y 1, y )(t 1 ) = y 1(t 1 )y (t 1 ) > 0 W (y 1, y )(t ) = y 1(t )y (t ) > 0
5 5 Hence y (t 1 ) < 0 and y (t ) > 0, so there is one zero of y. If W (y 1, y )(t) < 0 for all t. We get W (y 1, y )(t 1 ) = y 1(t 1 )y (t 1 ) < 0 W (y 1, y )(t ) = y 1(t )y (t ) < 0 Hence y (t 1 ) > 0 and y (t ) < 0, so there is one zero of y. We need to show there is only one zero of y between t 1 and t. If otherwise, then there is at least two zero of y between t 1 and t, we can select two consecutive zeros of y, say t 3 and t 4, such that t 1 < t 3 < t 1 < t 4 < t. Then from above proof we can show there is another zero of y 1, say t 5 such that t 1 < t 3 < t 1 < t 5 < t 4 < t. This contradict that t 1 and t are two consecutive zeros of y Euler Equation. An equation of the form t y + αty + βy = 0, t > 0, where α and β are real constants, is called an Euler equation. Show that the substitution x = ln t transform an Euler equation into an equation with constant coefficients. Answer: We get dx dt = 1 t d y dt = 1 d y t dx 1 dy t dx t y + αty + βy = d y + (α + 1)dy + βy = 0. dx dx
6 6 Section 3.5 Find the general solutions of the given differential equations. 1. y y + y = y + 4y + 9y = 0 Answer: 1. The characteristic equation is r r + 1 = (r 1)(r 1) = 0 Thus the possible value of r is r = 1, and the general solution of the equation is 8. The characteristic equation is y(t) = (c 1 + c t)e t. 16r + 4r + 90 Thus the possible value of r is r 1 = 3, and the general solution of the 4 equation is y(t) = (c 1 + c t)e 3 4 t. In each of the problems use the method of reduction of order to find second solution of the given equation. 3. t y 4ty + 6y = 0; t > 0; y 1 (t) = t 5. t y + 3ty + y = 0; t > 0; y 1 (t) = 1 t 6. t y t(t + )y + (t + )y = 0; t > 0; y 1 (t) = t
7 7 Answer: 3. Let y(t) = t v(t), then y = v t + tv y = v t + 4v t + v So we get. Then t 4 v = 0 v(t) = c 1 t + c and so y(t) = c 1 t 3 + c t. From y 1 (t) = t, we find the second solution is y (t) = t Let y(t) = t 1 v(t), then y = v t 1 t v y = t 1 v t v t + t 3 v So we get. Then tv + v = 0 v(t) = c 1 ln t + c and so y(t) = c 1 t 1 ln t + c t 1. From y 1 (t) = t 1, we find the second solution is y (t) = t 1 ln t. 6. Let y(t) = tv(t), then y = v t + v y = v t + v So we get. Then v v = 0 v(t) = c 1 e t + c
8 8 and so y(t) = c 1 te t + c t. From y 1 (t) = t, we find the second solution is y (t) = te t. 3. The differential equation y + δ(xy + y) = 0 arises in the study of the turbulent flow of a uniform stream past a circular cylinder. Verify that y 1 = exp( δx ) is one solution and then find the general solution in the form of an integral. Answer: Then it is easy to see y y 1 = e δx y 1 = δxe δx 1 = (δ x δ)e δx y 1 + δ(xy 1 + y 1 ) = 0. Let y = ve δx, then y = v e δx δxve δx y = v e δx δxv e δx + δ x ve δx δve δx. We get v δxv = 0, and then v(x) = x c 0 1e δx + c. Hence, x y(x) = (c 1 e δx + c )e δx. From y 1 = exp( δx ), we find the second solution is 0 y(x) = e δx x 0 e δx.
9 9 In the following problem use the method of of problem 33 to find second independent solution of the given equation. 34. t y + 3ty + y = 0; t > 0; y 1 (t) = 1 t Answer: The original equation can be written as From the Abel s theorem y + 3 t y + 1 t y = 0. We get and W (y 1, y ) = c 1 e 3dt t = c 1 t = y 1y 3 y 1y = y t + y t y + y t = c 1 t, y (t) = 1 t [c 1 ln t + c ]. 38. If a, b, c are positive constants, show that all solutions of ay + by + cy = 0 approach zero as t. Answer: The characteristic equation is Case A: b 4ac < 0 ar br + c = 0 Thus the possible values of r are r 1 = b+i 4ac b and r a = b i 4ac b, a and the general solution of the equation is y(t) = e bt 4ac b 4ac b a (c1 cos t + c sin t). a a
10 10 Hence, y(t) 0, as t, since a, b are positive constants. Case B: b 4ac = 0 Thus the possible values of r are r 1 = b, and the general solution of a the equation is y(t) = e bt a (c1 + c t). Hence, y(t) 0, as t, since a, b are positive constants. Case C: b 4ac > 0 Thus the possible values of r are r 1 = b+ b 4ac a and the general solution of the equation is y(t) = c 1 e b+ b 4ac a + c e b b 4ac a. and r = b b 4ac a, Hence, y(t) 0, as t, since a is positive constant and b + b 4ac < (a) If a > 0 and c > 0, but b = 0, show that the result of Problem 38 is no longer true, but that all solutions are bounded as t. (b) If a > 0 and b > 0, but c = 0, show that the result of Problem 38 is no longer true, but that all solutions approach a constant that depends on the initial conditions as t. Determine this constants for the initial conditions y(0) = y 0, y (0) = y ( 0). Answer: (a). a > 0 and b > 0, but c = 0 The characteristic equation is ar + cr = 0
11 Thus the possible values of r are r 1 = c i and r a = c i, and the a general solution of the equation is c c y(t) = c 1 cos a t + c sin a t. Hence, all solutions are bounded as t. 11 (b): a > 0 and b > 0, but c = 0 The characteristic equation is ar br = 0 Thus the possible values of r are r 1 = 0 and r = b, and the general a solution of the equation is y(t) = c 1 + c e bt a. Hence, y(t) c 1, as t, since a, b are positive constants. Obviously, c 1 depends on the initial conditions. From y (0) = y (0), we can get c = ay 0 b. From y(0) = y 0, we get c 1 = ay 0 b + y 0. Section 3.6 Answer: In each of the following problems, find the general solutions of the given differential equations. 1. y y + 3y = 3e t 4. y + y = sin t 7. y + 3y + y = t + 3 sin t 10. u + ω0u = cos ω 0 t 11. y + y + 4y = sinh t
12 1 Answer: 1. The characteristic equation is r r 3 = (r 3)(r + 1) = 0 Thus the possible values of r are r 1 = 3 and r = 1, and the general solution of the homogeneous equation is y(t) = c 1 e 3t + c e t. Let Y (t) = Ae t where A is a constant to be determined. On substituting to the original equation, we get 15Ae t = 3e t. So A = 1 5 and y(t) = c1e 3t + ce t 1 5 et is the general solution of the original equation. 4. The characteristic equation is r + r = r(r + ) = 0 Thus the possible values of r are r 1 = 0 and r =, and the general solution of the homogeneous equation is y(t) = c 1 + c e t. Let Y (t) = At + B sin t + C cos t where A, Band, C are constants to be determined. On substituting to the original equation, we get 4(B + C) sin t + 4(B C) cos t + A = sin t. So A = 3, B = 1 and C = 1 and y(t) = c 1 + c e t + 3t 1 (sin t + cos t) is the general solution of the original equation. 7. The characteristic equation is r + 3r + 1 = (r + 1)(r + 1) = 0
13 Thus the possible values of r are r 1 = 1 and r = 1, and the general solution of the homogeneous equation is y(t) = c 1 e t + c e t. Let Y (t) = A + Bt + Ct + D sin t + E cos t where A, B, C, D, ande are constants to be determined. On substituting to the original equation, we get 4C(A+3B)+(B+6C)t+Ct +( D 3E+D) sin t+( E+3D+E) cos t = t +3 sin t. 13 So A = 14, B = 6, C = 1, D = 3 10 and E = 9 13 and y(t) = c 1 e t + c e t + 14t 16Bt + t 3 10 sin t 9 13 cos t is the general solution of the original equation. 10. The characteristic equation is r + ω 0 = 0 Thus the possible values of r are r 1 = iω 0 and r = iω 0, and the general solution of the homogeneous equation is u(t) = c 1 cos ω 0 t + c sin ω 0 t. Let Y (t) = At cos ω 0 t + Bt sin ω 0 t where A and B are constants to be determined. On substituting to the original equation, we get So A = 0, B = 1 ω 0 Aω 0 sin ω 0 t + Bω 0 cos ω 0 t = cos ω 0 t. and u(t) = c 1 cos ω 0 t + c sin ω 0 t + is the general solution of the original equation. 11. The characteristic equation is r + r + 4 = 0 t ω 0 sin ω 0 t
14 14 Thus the possible values of r are r 1 = i and r = 1 15i, and the general solution of the homogeneous equation is u(t) = e t 15t 15t (c1 cos + c sin ). Let Y (t) = Ae t + Be t where A and B are constants to be determined. On substituting to the original equation, we get 6Ae t + 4Be t = e t + e t. So A = 1, B = 1 and 6 4 u(t) = e t 15t (c1 cos + c sin is the general solution of the original equation. 15t ) et 1 4 e t In each of the following problems, find the solutions of the given initial problem. 13. y + y y = t; y(0) = 0; y (0) = y y 3y = 3te t ; y(0) = 1; y (0) = 0 Answer: 13. The characteristic equation is r + r = 0 Thus the possible values of r are r 1 = and r = 1, and the general solution of the homogeneous equation is u(t) = c 1 e t + c e t. Let Y (t) = A + Bt where A and B are constants to be determined. On substituting to the original equation, we get B (A + Bt) = t.
15 15 So A = 1, B = 1 and y(t) = c 1 e t + c e t t 1 is the general solutions of the original equations. From y(0) = 0, we get c 1 + c = 1. From y (0) = 1, we get c 1 c = 1. Hence, c 1 = 1 and c = 1 and the solution of the initial problem is y(t) = e t 1 e t t The characteristic equation is r r 3 = 0 Thus the possible values of r are r 1 = 3 and r = 1, and the general solution of the homogeneous equation is u(t) = c 1 e 3t + c e t. Let Y (t) = (A+Bt)e t where A and B are constants to be determined. On substituting to the original equation, we get So A =, B = 1 and 3 B 3A 3Bt = 3t. y(t) = c 1 e t3 + c e t ( 3 t)et is the general solution of the original equation. From y(0) = 0, we get c 1 + c = 5 3. From y (0) = 1, we get 3c 1 c = 7 3. Hence, c 1 = 1 and c = 3 and the solution of the initial problem is y(t) = e 3t 3 e t (t + 3 )et.
16 16 Section 3.7 In each of Problems use the method of variation of parameters to find a particular solution of the differential equation. Then check your answer by using the method of undetermined coefficients.. y y y = e t 4. 4y 4y + y = 16e t Answer:. The characteristic equation is r r = 0 Thus the possible values of r are r 1 = and r = 1, and the general solutions of the homogeneous equation are u(t) = c 1 y 1 + c y = c 1 e t + c e t. is Method of variation of Parameters: W (y 1, y ) = 3e t then a particular solution of the original equation Hence, e Y (t) = e t t (e t ) dt + e t 3e t = 1 9 e t 3 te t e t (e t ) dt 3e t y(t) = c 1 e t + c e t 3 te t are the general solutions of the original equation. Method of undetermined coefficients: Let Y (t) = Ate t where A is constant to be determined. On substituting to the original equation, we get 3e t = e t.
17 17 So A = 3 and y(t) = c1e t + ce t 1 9 e t 3 te t is the general solution of the original equation. 4. The original equation can written as: The characteristic equation is y y y = 4e t r r = 0 Thus the possible values of r are r = 1, and the general solutions of the homogeneous equation are u(t) = c 1 y 1 + c y = c 1 e t + c te t. Method of variation of Parameters: W (y 1, y ) = e t then a particular solution of the original equation is t Y (t) = e t te (4e t t ) dt + te t e (4e t ) e t dt e t Hence, = t e t y(t) = c 1 e t + c te t + t e t are the general solutions of the original equation. Method of undetermined coefficients: Let Y (t) = At e t where A is constant to be determined. On substituting to the original equation, we get A = and y(t) = c 1 e t + c te t + t e t are the general solutions of the original equation.
18 18 In each of Problems find the general solution of the given differential equation. 5. y + y = tan t, 0 < t < π 8. y + 4y = 3 csc t, 0 < t < π Answer: 5.The characteristic equation is r + 1 = 0 Thus the possible values of r are r 1 = i and r = i, and the general solution of the equation is y(t) = c 1 cos t + c sin t. W (y 1, y ) = 1 then a particular solution of the original equation is Y (t) = cos t sin t tan tdt + sin t cos t tan tdt Hence, = cos t ln tan t + sec t, 0 < t < π y(t) = c 1 cos t + c sin t cos t ln tan t + sec t, 0 < t < π are the general solutions of the original equation. 8. The characteristic equation is r + 4 = 0 Thus the possible values of r are r 1 = i and r = i, and the general solution of the equation is y(t) = c 1 cos t + c sin t. W (y 1, y ) = then a particular solution of the original equation is 3 sin t csc t 3 cos t csc t Y (t) = cos t dt + sin t dt = 3t cos t sin t ln sin t, 0 < t < π
19 19 Hence, y(t) = c 1 cos t + c sin t 3t cos t sin t ln sin t, 0 < t < π are the general solutions of the original equation. In each of the problems verify that the given functions y 1 and y satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous equation. 13. t y y = 3t 1, t > 0; y 1 = t, y = t ty + (1 + t)y + 4y = t e t, y 1 = 1 + t, y = e t Answer: 5. It is easy to check that y 1 and y satisfy W (y 1, y ) = 3 and let t y y = 3t 1, t > 0. Y (t) = t t 1 (3t 1) 3 + t 1 t (3t 1) 3 = 3 10 t4 1 3 t ln t t, t > 0. then a particular solution of the original equation is y(t) = 3 10 t4 1 3 t ln t, t > It is easy to check that y 1 and y satisfy ty + (1 + t)y + 4y = t e t W (y 1, y ) = te t and let e t (t e t ) Y (t) = (1 + t) te t = ( 1 t 5 4 t )et. (1 + t)(t + e t e t ) te t
20 0 then a particular solution of the original equation is y(t) = ( 1 t 5 4 t )et.. By choosing the lower limit of the integration in Equ.(8) in the text as the initial point, show that Y (t) becomes Y (t) = t y 1 (s)y (t) y 1 (t)y (s) y 1 (s)y (s) y 1(s)y (s) g(s)ds. Show that Y (t) is a solution of the initial value problem L[y] = g(t), y( ) = 0, y ( ) = 0. Thus Y can be identified with v in problem 1. Answer: Y (t) = y 1 (t) y (s)g(s) W (y 1, y )(s) ds + y (t) y 1 (s)g(s) W (y 1, y )(s) ds = t y 1 (t)y (s)g(s) t y 1 (s)y (s) y (s)y 1(s) ds + y (t)y 1 (s)g(s) y 1 (s)y (s) y (s)y 1(s) ds Hence, = t Y (0) = y 1 (s)y (t) y (s)y 1 (t) y 1 (s)y (s) y (s)y 1(s) g(s)ds t0 y 1 (s)y (t) y (s)y 1 (t) y 1 (s)y (s) y (s)y 1(s) g(s)ds From Y (t) = y 1 (t) t y (s)g(s) t W (y 1, y )(s) ds + y y 1 (s)g(s) (t) W (y 1, y )(s) ds,
21 we get then Y (t) = y 1(t) Y ( ) = y 1( ) t0 t y (s)g(s) t W (y 1, y )(s) ds + y 1 (s)g(s) y (t) W (y 1, y )(s) ds y (s)g(s) t0 W (y 1, y )(s) ds + y ( ) t y Y (t) = y (s)g(s) 1 (t) ds + y (t) W (y 1, y )(s) Hence, Y (t) + p(t)y (t) + q(t)y (t) t 1 y 1 (s)g(s) W (y 1, y )(s) ds = 0 y 1 (s)g(s) ds + g(t) W (y 1, y )(s) t y = [y (t) + p(t)y (t) 1 (s)g(s) + q(t)y (t)] W (y 1, y )(s) ds t y [y 1 (t) + p(t)y 1(t) (s)g(s) + q(t)y 1 (t)] ds + g(t) g(t). W (y 1, y )(s) Therefore, Y (t) is a solution of the initial value problem L[y] = g(t), y( ) = 0, y ( ) = (a) Use the result of Problem to show that the solution of the initial value problem y + y = g(t), y( ) = 0, y ( ) = 0 is y = t sin (t s)g(s)ds (b) Find the solution of the initial value problem y + y = g(t), y(0) = y 0, y (0) = y 0.
22 Answer: The characteristic equation is r + 1 = 0 Thus the possible values of r are r 1 = i and r = i, and the general solution of the homogeneous equation is y(t) = c 1 cos t + c sin t. (a) From the result in Problem, we get the solution of the initial problem Y (t) = t y 1 (s)y (t) y 1 (t)y (s) y 1 (s)y (s) y 1(s)y (s) g(s)ds (b) Because = = t t cos s sin t cos t sin s cos s cos s + sin s sin s g(s)ds (cos s sin t cos t sin s)g(s)ds = t sin (t s)g(s)ds y(t) = c 1 cos t + c sin t satisfy the corresponding homogenous equation, let y(0) = y 0 and y (0) = y 0, then we get c 1 = y 0 and c = y 0. Hence, the solution of the initial problem is y(t) = y 0 cos t + y 0 sin t + t sin (t s)g(s)ds. 8. The method of reduction of order (Section 3.5) can also be used for the nonhomogeneous equation y (t) + p(t)y (t) + q(t)y(t) = g(t), (i)
23 provided one solution y 1 of the corresponding homogeneous equation is known. Let y = v(t)y 1 (t) and show that y satisfies Equ.(i) if and only if y 1 v + [y 1(t) + p(t)y 1 (t)]v = g(t), (ii) Equation (ii) is a first order linear equation for v. Solving this equation, integrating the result, and then multiplying by y 1 (t) lead to the general solution of Eq.(i). 3 Answer: Let y = v(t)y 1 (t), then So, y (t) = v (t)y 1 (t) + v(t)y 1(t) y (t) = v (t)y 1 (t) + v (t)y 1(t) + v(t)y 1 (t) y (t) + p(t)y (t) + q(t)y(t) = y 1 v + [y 1(t) + p(t)y 1 (t)]v + v(y 1 (t) + p(t)y 1(t) + q(t)y 1 (t)) = g(t) if y 1 is the solution of the corresponding homogeneous equation and y 1 v + [y 1(t) + p(t)y 1 (t)]v = g(t). In each of the problems use the method outlined in Problem 8 to solve the given differential equation. 9. t y ty + y = 4t, t > 0; y 1 (t) = t 30. 4y + 7ty + 5y = t, t > 0; y 1 (t) = t Answer: 9. Use the method in Problem 8, the original equation can be written as y (t) t y (t) + y(t) = 4. t
24 4 Let v satisfies then tv = 4 and v = 4t ln t + c 1 t. y 1 v + [y 1(t) + p(t)y 1 (t)]v = g(t) So y = y 1 v = 4t ln t+c 1 t is the solution of t y ty +y = 4t, t > 0. Hence, the general solution are y = c 1 t + c t + 4t ln t 9. Use the method in Problem 8, the original equation can be written as Let v satisfies y (t) + 7 t y (t) + 5 t y(t) = 1 t. y 1 v + [y 1(t) + p(t)y 1 (t)]v = g(t) then v + 5 t v = 1 and v = 1 1 t + c 1 t 4. So y = y 1 v = 1 1 t + c 1t 5 is the solution of 4y + 7ty + 5y = t, t > 0. Hence, the general solution are y = c 1 t 5 + c t t. Section 3.8 In the following problem, determine ω 0, R and δ so as to write the given expression in the form u = R cos ω 0 t δ. 3. u = 4 cos 3t sin 3t
25 Answer: R = 4 + ( ) = 5 and δ = arctan( 4 ) == Hence u = 5 cos 3t δ with δ = arctan( 4 ) == A mass weighing lb stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position u of the mass at time t. Plot u versus t. Find the frequency, period, and amplitude of the motion. Answer: Generally, the motion the mass is described by the following equation mu (t) + γu (t) + ku(t) = F (t). Since nothing is said in the statement of the problem about an external force, we assume that F (t) = 0. Also, γ = 0 because there is no damping. To determine m note that m = w g = lb 3ft sec = 1 16 lb sec. ft The spring constant k is be found from the statement that the mass stretches the spring 6 in, or by 1 k = ft. Thus lb 1/ft = 4 lb ft. The equation of motion of the mass is u + 64u(t) = 0. The initial conditions are u(0) = 1 4 and u (0) = 0. The general solution is u = A cos(8t) + B sin(8t).
26 6 The solution satisfies the initial conditions u(0) = 1 4 and u = 0, so we can get A = 1 and B = 0. 4 Hence, the position u of the mass at time t is 1 cos(8t)ft, tin sec; ω = 8rad/sec, T = π/4sec, R = 1/4ft Assume that the system described by the equation mu +γu +ku = 0 is either critically damped or overdamped. Show that the mass can pass through the equilibrium position at most once, regardless of the initial conditions. Answer: Case A: If the system is critically damped, then γ = km. The characteristic equation of the original differential equation is mr + γr + k = 0 Thus the possible value of r is r = γ, and the general solution of the m homogeneous equation is y(t) = (c 1 + c t)e γ m t. Obviously, y(t) at most have one zero, regardless the coefficients of c 1 and c, because y(t) always nondecreasing or nonincreasing. Hence, the mass can pass through the equilibrium position at most once, regardless of the initial conditions. Case B: If the system is overdamped, then γ > km. The characteristic equation of the original differential equation is mr + γr + k = 0
27 γ 4mk m Thus the possible values of r are r 1 = γ+ γ 4mk < 0 and r m = γ < 0 for γ > km, and the general solution of the homogeneous equation is y(t) = c 1 e γ+ = e γ m (c1 e γ 4mk m γ 4mk m t + c e γ γ 4mk t m t + c e γ 4mk m t ). 7 Obviously, y(t) at most have one zero, regardless the coefficients of c 1 and c, then the mass can pass through the equilibrium position at most once, regardless of the initial conditions. 4. The position of a certain spring-mass system satisfied the initial value problem 3 u + ku = 0, u(0) =, u (0) = v. If the period and amplitude of the resulting motion are observed to be π and 3, respectively, determine the value of k and v. Answer: The period of the motion is T = π( m k ) 1 = π( 3/ k ) 1 = π. So we get k = 6 and the equation can written as u + 4u = 0. Obviously, the general solution of this equation is From u(0) =, then A =. u(t) = A cos t + B sin t. From the amplitude of the resulting motion is 3, R = A + B = 3 and then B = ± 5. Hence, u(t) = cos t + ± 5 sin t
28 8 and v = u (0) = ± 5 Section Consider a vibration system described by the initial value problem u u + u = cos ωt, u(0) = 0, u (0) =. (a.) Determine the steady-state part of the solution of this problem. (b.) Find the amplitude A of the steady-state solution in terms of ω. (c.) Plot A versus ω. (d.) Find the maximum value of A and the frequency ω for which it occurs. Answer: The characteristic equation of the original differential equation is r r + = 0 Thus the possible values of r are r 1 = i r 8 8 = 1 17i, and 8 8 the general solution of the homogeneous equation is u(t) = e t (c 1 cos t + c sin t). 8 8 The motion of this system can be described by u(t) = e t (c 1 cos t + c sin t) + R cos(ωt δ). 8 8 where R = / ( ω ) + ω 16
29 and Figure 1. for problem 17 ω = 0.7 cos δ = sin δ = ω ( ω ) + ω 16 1 ω 4 ( ω ) + ω 16 (a.) The steady-state part of the solution of this problem is u = [3( ω ) cos ωt + 8ω sin ωt] 64 63ω + 16ω 4. (b.)the amplitude A of the steady-state is A = ω + 16ω 4. (c.) For graph of A versus ω, see Figure 1. (d.)the maximum value of A is A = 64 17
30 30 and the corresponding frequency ω = Consider the forced but undamped system described by the initial value problem u + u = 3 cos ωt, u(0) = 0, u (0) = 0. a. Find the solution u(t) for ω 1. b. Plot the solution u(t) versus t for ω = 0.7, ω = 0.8 and ω = 0.9. Describe how the response u(t) changes as ω varies in this interval. What happens as ω takes on values closer and closer to 1? Note that the natural frequency of the unforced system is ω 0 = 1. Answer: (a.) ω 0 = k/m = 1, if ω ω 0, then the general solution is u = c 1 cos t + c sin t + 3 cos ωt. 1 ω From u(0) = 0, u (0) = 0, we get c 1 = 3 1 ω, c = 0. Hence the solution is u(t) = 3 (cos ωt cos t). 1 ω (b.) For the graphs of the solution u(t) versus t for ω = 0.7, ω = 0.8 and ω = 0.9, see Figure., Figure 3. and Figure 4. We can write above solution as: 3 (1 ω)t (1 + ω)t u(t) = ( sin ) sin. 1 ω If 1 ω is small, the 1 + ω is much greater than 1 ω. Consequently, sin (1+ω)t is rapidly oscillating function compared to sin (1 ω)t. Thus the motion is a rapid oscillation with frequency 1+ω but with a
31 Figure. for problem 18 ω = 0.7 slowly varying sinusoidal amplitude 3 1 ω sin (1 ω)t. The amplitude of u(t) gets larger and lager as w varies from ω = 0.7, ω = 0.8 to ω = 0.9, and closer and closer to 1, the natural frequency of the unforced system.
32 Figure 3. for problem 18 ω = 0.8; Figure 4. for problem 18 ω = 0.9
Lecture Notes for Math250: Ordinary Differential Equations
Lecture Notes for Math250: Ordinary Differential Equations Wen Shen 2011 NB! These notes are used by myself. They are provided to students as a supplement to the textbook. They can not substitute the textbook.
More informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More informationSo far, we have looked at homogeneous equations
Chapter 3.6: equations Non-homogeneous 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 informationApplications of Second-Order Differential Equations
Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationSECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
More informationMath 22B, Homework #8 1. y 5y + 6y = 2e t
Math 22B, Homework #8 3.7 Problem # We find a particular olution of the ODE y 5y + 6y 2e t uing the method of variation of parameter and then verify the olution uing the method of undetermined coefficient.
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
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 closed-book test, and is to be taken without the use
More informationGeneral Theory of Differential Equations Sections 2.8, 3.1-3.2, 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, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationSecond-Order Linear Differential Equations
Second-Order Linear Differential Equations A second-order linear differential equation has the form 1 Px d 2 y dx 2 dy Qx dx Rxy Gx where P, Q, R, and G are continuous functions. We saw in Section 7.1
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationA Brief Review of Elementary Ordinary Differential Equations
1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
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 informationNonhomogeneous Linear Equations
Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where
More informationSecond Order Linear Differential Equations
CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution
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 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 informationHow To Solve A Linear Dierential Equation
Dierential Equations (part 2): Linear Dierential Equations (by Evan Dummit, 2012, v. 1.00) Contents 4 Linear Dierential Equations 1 4.1 Terminology.................................................. 1 4.2
More information19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.
More informationDifferential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University
Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter. Linear second order ODEs 5.. Newton s second law 5.2. Springs
More informationSystem of First Order Differential Equations
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
More information9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients
September 29, 201 9-1 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential
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 informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationStructural Dynamics, Dynamic Force and Dynamic System
Structural Dynamics, Dynamic Force and Dynamic System Structural Dynamics Conventional structural analysis is based on the concept of statics, which can be derived from Newton s 1 st law of motion. This
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationtegrals as General & Particular Solutions
tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx
More informationHomework #2 Solutions
MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution
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 information1. (from Stewart, page 586) Solve the initial value problem.
. (from Stewart, page 586) Solve the initial value problem.. (from Stewart, page 586) (a) Solve y = y. du dt = t + sec t u (b) Solve y = y, y(0) = 0., u(0) = 5. (c) Solve y = y, y(0) = if possible. 3.
More informationMethods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College
Methods of Solution of Selected Differential Equations Carol A. Edwards Chandler-Gilbert Community College Equations of Order One: Mdx + Ndy = 0 1. Separate variables. 2. M, N homogeneous of same degree:
More informationPart IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3. Stability and pole locations.
Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3 Stability and pole locations asymptotically stable marginally stable unstable Imag(s) repeated poles +
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationLet s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1.
Examples of Transient and RL Circuits. The Series RLC Circuit Impulse response of Circuit. Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure.
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point
More informationMath 267 - Practice exam 2 - solutions
C Roettger, Fall 13 Math 267 - Practice exam 2 - solutions Problem 1 A solution of 10% perchlorate in water flows at a rate of 8 L/min into a tank holding 200L pure water. The solution is kept well stirred
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More informationReview of First- and Second-Order System Response 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.151 Advanced System Dynamics and Control Review of First- and Second-Order System Response 1 1 First-Order Linear System Transient
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 informationSOLUTIONS TO CONCEPTS CHAPTER 15
SOLUTIONS TO CONCEPTS CHAPTER 15 1. v = 40 cm/sec As velocity of a wave is constant location of maximum after 5 sec = 40 5 = 00 cm along negative x-axis. [(x / a) (t / T)]. Given y = Ae a) [A] = [M 0 L
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationLecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
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 informationUnit - 6 Vibrations of Two Degree of Freedom Systems
Unit - 6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two
More informationCollege of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions
College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use
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 informationStep response of an RLC series circuit
School of Engineering Department of Electrical and Computer Engineering 332:224 Principles of Electrical Engineering II Laboratory Experiment 5 Step response of an RLC series circuit 1 Introduction Objectives
More informationMath 2280 - Assignment 6
Math 2280 - Assignment 6 Dylan Zwick Spring 2014 Section 3.8-1, 3, 5, 8, 13 Section 4.1-1, 2, 13, 15, 22 Section 4.2-1, 10, 19, 28 1 Section 3.8 - Endpoint Problems and Eigenvalues 3.8.1 For the eigenvalue
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More informationSecond Order Systems
Second Order Systems Second Order Equations Standard Form G () s = τ s K + ζτs + 1 K = Gain τ = Natural Period of Oscillation ζ = Damping Factor (zeta) Note: this has to be 1.0!!! Corresponding Differential
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 re-writing F(s) in a form suitable for the use of the
More informationSeries FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.
More informationSensor Performance Metrics
Sensor Performance Metrics Michael Todd Professor and Vice Chair Dept. of Structural Engineering University of California, San Diego mdtodd@ucsd.edu Email me if you want a copy. Outline Sensors as dynamic
More informationNumerical Solution of Differential Equations
Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant
More information12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*
+ v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure
More information1. First-order Ordinary Differential Equations
Advanced Engineering Mathematics 1. First-order ODEs 1 1. First-order Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential
More informationdy dx and so we can rewrite the equation as If we now integrate both sides of this equation, we get xy x 2 C Integrating both sides, we would have
LINEAR DIFFERENTIAL EQUATIONS A first-der linear differential equation is one that can be put into the fm 1 d Py Q where P and Q are continuous functions on a given interval. This type of equation occurs
More informationSeparable First Order Differential Equations
Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationy cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C
Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.
More informationSolutions to Linear First Order ODE s
First Order Linear Equations In the previous session we learned that a first order linear inhomogeneous ODE for the unknown function x = x(t), has the standard form x + p(t)x = q(t) () (To be precise we
More informationCoffeyville Community College #MATH 202 COURSE SYLLABUS FOR DIFFERENTIAL EQUATIONS. Ryan Willis Instructor
Coffeyville Community College #MATH 202 COURSE SYLLABUS FOR DIFFERENTIAL EQUATIONS Ryan Willis Instructor COURSE NUMBER: MATH 202 COURSE TITLE: Differential Equations CREDIT HOURS: 3 INSTRUCTOR: OFFICE
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationLesson 11. Luis Anchordoqui. Physics 168. Tuesday, December 8, 15
Lesson 11 Physics 168 1 Oscillations and Waves 2 Simple harmonic motion If an object vibrates or oscillates back and forth over same path each cycle taking same amount of time motion is called periodic
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 informationPhysics 231 Lecture 15
Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationSimple Harmonic Motion Experiment. 1 f
Simple Harmonic Motion Experiment In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More information1.2 Second-order systems
1.2. SECOND-ORDER SYSTEMS 25 if the initial fluid height is defined as h() = h, then the fluid height as a function of time varies as h(t) = h e tρg/ra [m]. (1.31) 1.2 Second-order systems In the previous
More informationAP Physics C. Oscillations/SHM Review Packet
AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationRecognizing Types of First Order Differential Equations E. L. Lady
Recognizing Types of First Order Differential Equations E. L. Lady Every first order differential equation to be considered here can be written can be written in the form P (x, y)+q(x, y)y =0. This means
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationCOWLEY COLLEGE & Area Vocational Technical School
COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR DIFFERENTIAL EQUATIONS MTH 4465 3 Credit Hours Student Level: This course is open to students on the college level in the sophomore
More informationNotice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW1 Possible Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 14.P.003 An object attached to a spring has simple
More informationcorrect-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More information2.2 Magic with complex exponentials
2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or
More informationAP Calculus BC 2001 Free-Response Questions
AP Calculus BC 001 Free-Response Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
More informationHomework #1 Solutions
MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : ANALYTICAL METHODS FOR ENGINEERS Unit code: A/60/40 QCF Level: 4 Credit value: 5 OUTCOME 3 - CALCULUS TUTORIAL DIFFERENTIATION 3 Be able to analyse and model engineering situations and solve problems
More informationEngineering Feasibility Study: Vehicle Shock Absorption System
Engineering Feasibility Study: Vehicle Shock Absorption System Neil R. Kennedy AME40463 Senior Design February 28, 2008 1 Abstract The purpose of this study is to explore the possibilities for the springs
More informationChapter 4. Linear Second Order Equations. ay + by + cy = 0, (1) where a, b, c are constants. The associated auxiliary equation is., r 2 = b b 2 4ac 2a
Chapter 4. Linear Second Order Equations ay + by + cy = 0, (1) where a, b, c are constants. ar 2 + br + c = 0. (2) Consequently, y = e rx is a solution to (1) if an only if r satisfies (2). So, the equation
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
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 information2.2 Separable Equations
2.2 Separable Equations 73 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More information3. Which of the following couldn t be the solution of a differential equation? (a) z(t) = 6
MathQuest: Differential Equations What is a Differential Equation? 1. Which of the following is not a differential equation? (a) y = 3y (b) 2x 2 y + y 2 = 6 (c) tx dx dt = 2 (d) d2 y + 4 dy dx 2 dx + 7y
More informationStructural Dynamics of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems
Structural Dynamics of Linear Elastic Single-Degree-of-Freedom (SDOF) Systems SDOF Dynamics 3-1 This set of slides covers the fundamental concepts of structural dynamics of linear elastic single-degree-of-freedom
More informationTechniques of Integration
CHPTER 7 Techniques of Integration 7.. Substitution Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration
More informationReview of Fourier series formulas. Representation of nonperiodic functions. ECE 3640 Lecture 5 Fourier Transforms and their properties
ECE 3640 Lecture 5 Fourier Transforms and their properties Objective: To learn about Fourier transforms, which are a representation of nonperiodic functions in terms of trigonometric functions. Also, to
More information