# ORDINARY DIFFERENTIAL EQUATIONS

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1 ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations. We describe the main ideas to solve certain differential equations, like first order scalar equations, second order linear equations, and systems of linear equations. We use power series methods to solve variable coefficients second order linear equations. We introduce Laplace transform methods to find solutions to constant coefficients equations with generalized source functions. We provide a brief introduction to boundary value problems, Sturm-Liouville problems, and Fourier Series expansions. We end these notes solving our first partial differential equation, the Heat Equation. We use the method of separation of variables, hence solutions to the partial differential equation are obtained solving infinitely many ordinary differential equations. x 2 x 2 x b a x

2 G. NAGY ODE September 4, 25 I Contents Chapter. First Order Equations.. Linear Constant Coefficient Equations 2... Overview of Differential Equations Linear Equations Linear Equations with Constant Coefficients The Initial Value Problem Exercises 9.2. Linear Variable Coefficient Equations.2.. Linear Equations with Variable Coefficients.2.2. The Initial Value Problem The Bernoulli Equation Exercises 8.3. Separable Equations Separable Equations Euler Homogeneous Equations Exercises Exact Equations Exact Differential Equations Finding a Potential Function The Integrating Factor Method The Integrating Factor for the Inverse Function Exercises 4.5. Applications Exponential Decay Newton s Cooling Law Salt in a Water Tank Exercises 5.6. Nonlinear Equations The Picard-Lindelöf Theorem Comparison Linear Nonlinear Equations Direction Fields Exercises 63 Chapter 2. Second Order Linear Equations Variable Coefficients The Initial Value Problem Homogeneous Equations The Wronskian Function Abel s Theorem Exercises Reduction of Order Methods Special Second Order Equations Reduction of Order Method Exercises Homogeneous Constant Coefficients Equations The Roots of the Characteristic Polynomial Real Solutions for Complex Roots Constructive proof of Theorem

3 II G. NAGY ODE september 4, Exercises Nonhomogeneous Equations The General Solution Formula The Undetermined Coefficients Method The Variation of Parameters Method Exercises Applications Review of Constant Coefficient Equations Undamped Mechanical Oscillations Damped Mechanical Oscillations Electrical Oscillations Exercises Chapter 3. Power Series Solutions Solutions Near Regular Points Regular Points The Power Series Method The Legendre Equation Exercises The Euler Equidimensional Equation The Roots of the Indicial Polynomial Real Solutions for Complex Roots Transformation to Constant Coefficients Exercises Solutions Near Regular Singular Points Regular Singular Points The Frobenius Method The Bessel Equation Exercises 46 Notes on Chapter 3 47 Chapter 4. The Laplace Transform Method Definition of the Laplace Transform Review of Improper Integrals Definition and Table Main Properties Exercises The Initial Value Problem Solving Differential Equations One-to-One Property Partial Fractions Exercises Discontinuous Sources Step Functions Translation Identities Solving Differential Equations Exercises Generalized Sources Sequence of Functions and the Dirac Delta Computations with the Dirac Delta 82

4 G. NAGY ODE September 4, 25 III Applications of the Dirac Delta The Impulse Response Function Comments on Generalized Sources Exercises Convolutions and Solutions Definition and Properties The Laplace Transform Solution Decomposition Exercises 96 Chapter 5. Systems of Differential Equations Linear Differential Systems First Order Linear Systems Order Transformations The Initial Value Problem Homogeneous Systems The Wronskian and Abel s Theorem Exercises Constant Coefficients Diagonalizable Systems Decoupling the systems Eigenvector solution formulas Alternative solution formulas Non-homogeneous systems Exercises Two-by-Two Constant Coefficients Systems The diagonalizable case Non-diagonalizable systems Exercises Two-by-Two Phase Portraits Real distinct eigenvalues Complex eigenvalues Repeated eigenvalues Exercises Non-Diagonalizable Systems 24 Chapter 6. Autonomous Systems and Stability Flows on the Line Autonomous Equations A Geometric Analysis Population Growth Models Linear Stability Analysis Exercises Flows on the Plane Linear Stability 255 Chapter 7. Boundary Value Problems Eigenvalue-Eigenfunction Problems Comparison: IVP and BVP Eigenvalue-eigenfunction problems Exercises 265

5 IV G. NAGY ODE september 4, Overview of Fourier Series Origins of Fourier series Fourier series Even and odd functions Exercises The Heat Equation The Initial-Boundary Value Problem The Separation of Variables Exercises 28 Chapter 8. Review of Linear Algebra Systems of Algebraic Equations Linear algebraic systems Gauss elimination operations Linearly dependence Exercises Matrix Algebra A matrix is a function Matrix operations The inverse matrix Determinants Exercises Diagonalizable Matrices Eigenvalues and eigenvectors Diagonalizable matrices The exponential of a matrix Exercises 34 Chapter 9. Appendices 35 Appendix A. Review Complex Numbers 35 Appendix B. Review of Power Series 36 Appendix C. Review Exercises 32 Appendix D. Practice Exams 32 Appendix E. Answers to Exercises 32 References 33

6 G. NAGY ODE September 4, 25 Chapter. First Order Equations We start our study of differential equations in the same way the pioneers in this field did. We show particular techniques to solve particular types of first order differential equations. The techniques were developed in the eighteen and nineteen centuries and the equations include linear equations, separable equations, Euler homogeneous equations, and exact equations. Soon this way of studying differential equations reached a dead end. Most of the differential equations cannot be solved by any of the techniques presented in the first sections of this chapter. People then tried something different. Instead of solving the equations they tried to show whether an equation has solutions or not, and what properties such solution may have. This is less information than obtaining the solution, but it is still valuable information. The results of these efforts are shown in the last sections of this chapter. We present Theorems describing the existence and uniqueness of solutions to a wide class of differential equations. y y = 2 cos(t) cos(y) π 2 t π 2

7 2 G. NAGY ODE september 4, 25.. Linear Constant Coefficient Equations... Overview of Differential Equations. A differential equation is an equation, the unknown is a function, and both the function and its derivatives may appear in the equation. Differential equations are essential for a mathematical description of nature, because they are the central part many physical theories. A few examples are Newton s and Lagrange equations for classical mechanics, Maxwell s equations for classical electromagnetism, Schrödinger s equation for quantum mechanics, and Einstein s equation for the general theory of gravitation. In the following examples we show how differential equations look like. (a) Newton s Law: ma = f, mass times acceleration equals force. Newton s second law of motion for a single particle is a differential equation. The unknown is the position of the particle in space, x(t), at the time t. This is a single variable vector-valued function in space. In another words, x : R R 3, where x is a function with domain R and range R 3. The differential equation is m d2 x (t) = f (t, x(t)), dt2 where the positive constant m is the mass of the particle, d 2 x/dt 2 is the acceleration of the particle, and f : R R 3 R 3 is the force acting on the particle, which depends on the time t and the position in space x. (b) Radioactive Decay: In the time decay of a radioactive substance the unknown is a scalar-valued function u : R R, where u(t) is the concentration of the radioactive substance at the time t. The differential equation is du (t) = k u(t), dt where k is a positive constant. The equation says the higher the material concentration the faster it decays. (c) The Wave Equation: The wave equation describes waves propagating in a media. An example is sound, where pressure waves propagate in the air. The unknown is a scalarvalued function of two variables u : R R 3 R, where u(t, x) is a perturbation in the air density at the time t and point x = (x, y, z) in space. (We used the same notation for vectors and points, although they are different type of objects.) The equation is tt u(t, x) = v 2[ xx u(t, x) + yy u(t, x) + zz u(t, x) ], where v is a positive constant describing the wave speed, and we have used the notation to mean partial derivative. (d) The Heat Equation: The heat equation describes the change of temperature in a solid material as function of time and space. The unknown is a scalar-valued function u : R R 3 R, where u(t, x) is the temperature at time t and the point x = (x, y, z) in the solid. The equation is t u(t, x) = k [ xx u(t, x) + yy u(t, x) + zz u(t, x) ], where k is a positive constant representing thermal properties of the material. The equations in examples (a) and (b) are called ordinary differential equations (ODE), since the unknown function depends on a single independent variable, t in these examples. The equations in examples (c) and (d) are called partial differential equations (PDE), since the unknown function depends on two or more independent variables, t, x, y, and z in these examples, and their partial derivatives appear in the equations. The order of a differential equation is the highest derivative order that appears in the equation. Newton s equation in Example (a) is second order, the time decay equation in Example (b) is first order, the wave equation in Example (c) is second order is time and

8 G. NAGY ODE September 4, 25 3 space variables, and the heat equation in Example (d) is first order in time and second order in space variables...2. Linear Equations. We start with a precise definition of the differential equations we study in this Chapter. We use primes to denote derivatives, dy dt (t) = y (t), because it is a compact notation. We use it when there is no risk of confusion. Definition... A first order ordinary differential equation in the unknown y is y (t) = f(t, y(t)), (..) where y : R R is the unknown function and f : R 2 R is a given function. The equation in (..) is called linear iff the function with values f(t, y) is linear on its second argument; that is, there exist functions a, b : R R such that y (t) = a(t) y(t) + b(t), f(t, y) = a(t) y + b(t). (..2) A linear first order equation has constant coefficients iff both functions a and b in Eq. (..2) are constants. Otherwise, the equation has variable coefficients. A different sign convention for Eq. (..2) may be found in the literature. For example, Boyce-DiPrima [3] writes it as y = a y + b. The sign choice in front of function a is just a convention. Some people like the negative sign, because later on, when they write the equation as y + a y = b, they get a plus sign on the left-hand side. In any case, we stick here to the convention y = ay + b. Example..: (a) An example of a first order linear ODE is the equation y (t) = 2 y(t) + 3. In this case, the right-hand side is given by the function f(t, y) = 2y + 3, where we can see that a(t) = 2 and b(t) = 3. Since these coefficients do not depend on t, this is a constant coefficient equation. (b) Another example of a first order linear ODE is the equation y (t) = 2 y(t) + 4t. t In this case, the right hand side is given by the function f(t, y) = 2y/t + 4t, where a(t) = 2/t and b(t) = 4t. Since the coefficients are nonconstant functions of t, this is a variable coefficients equation. A function y : D R R is solution of the differential equation in (..) iff the equation is satisfied for all values of the independent variable t in the domain D of the function y. Example..2: Show that y(t) = e 2t 3 2 is solution of the equation y (t) = 2 y(t) + 3. Solution: We need to compute the left and right-hand sides of the equation and verify they agree. On the one hand we compute y (t) = 2e 2t. On the other hand we compute ( 2 y(t) + 3 = 2 e 2t 3 ) + 3 = 2e 2t. 2 We conclude that y (t) = 2 y(t) + 3 for all t R.

9 4 G. NAGY ODE september 4, 25 Example..3: Find the differential equation y = f(t, y) satisfied by y(t) = 4 e 2t + 3. Solution: We compute the derivative of y, y = 8 e 2t We now write the right hand side above, in terms of the original function y, that is, y = 4 e 2t + 3 y 3 = 4 e 2t 2(y 3) = 8 e 2t. So we got a differential equation satisfied by y, namely y = 2y Linear Equations with Constant Coefficients. Constant coefficient equations are simpler to solve than variable coefficient ones. There are many ways to solve them. Integrating each side of the equation, however, does not work. For example, take the equation and integrate on both sides, y (t) dt = 2 y = 2 y + 3, y(t) dt + 3t + c, c R. The Fundamental Theorem of Calculus implies y(t) = y (t) dt. Using this equality in the equation above we get y(t) = 2 y(t) dt + 3t + c. Integrating both sides of the differential equation is not enough to find a solution y. We still need to find a primitive of y. Since we do not know y, we cannot find its primitive. We have only rewritten the original differential equation as an integral equation. So, integrating both sides of a linear equation does not work. One needs a better idea to solve the equation. We describe here one way, the integrating factor method. Multiply the differential equation by a function µ, µ y = µ(a y + b), µ is called an integrating factor. Choose an integrating factor having one important property. The differential equation is transformed into a total derivative, y = ay + b dψ (t, y(t)) =, dt that is, a total derivative of a function ψ. This function depends on t and y and is called a potential function. Integrating the differential equation is now trivial, the potential function must be a constant, ψ(t, y(t)) = c. A solution y of the differential equation is given implicitly by the equation ψ(t, y(t)) = c. This is the main idea of the integrating factor method. The difficult part is to find an appropriate integrating factor function µ. We now state in a theorem a precise formula for the solutions of constant coefficient linear equations. Theorem..2 (Constant Coefficients). The linear differential equation y (t) = a y(t) + b (..3) where a, b are constants, has infinitely many solutions labeled by c R as follows, y(t) = c e at b a. (..4)

10 G. NAGY ODE September 4, 25 5 Remarks: (a) Eq. (..4) is called the general solution of the differential equation in (..3). Theorem..2 says that Eq. (..3) has infinitely many solutions, one solution for each value of the constant c, which is not determined by the equation. This is reasonable. Since the differential equation contains one derivative of the unknown function y, finding a solution of the differential equation requires to compute an integral. Every indefinite integration introduces an integration constant. This is the origin of the constant c above. (b) In the next Section we generalize this idea to find solutions of linear equations with variable coefficients. In.4 we generalize this idea to certain nonlinear differential equations. Proof of Theorem..2: Write the equation with y on one side only, y a y = b, and then multiply the differential equation by a function µ, called an integrating factor, Now comes the critical step. We choose a function µ such that µ y a µ y = µ b. (..5) a µ = µ. (..6) For any function µ solution of Eq. (..6), the differential equation in (..5) has the form µ y + µ y = µ b. But the left-hand side is a total derivative of a product of two functions, ( µ y ) = µ b. (..7) This is the property we want in an integrating factor, µ. We want to find a function µ such that the left-hand side of the differential equation for y can be written as a total derivative, just as in Eq. (..7). We only need to find one of such functions µ. So we go back to Eq. (..6), the differential equation for µ, which is simple to solve, ( µ = a µ µ ln(µ)) µ = a = a ln(µ) = at + c. Computing the exponential of both sides in the equation above we get µ = e at+c = e at e c µ = c e at, c = e c. Since c is a constant which will cancel out from Eq. (..5) anyway, we choose the integration constant c =, hence c =. The integrating function is then µ(t) = e at. This function is an integrating factor, because if we start again at Eq. (..5), we get e at y a e at y = b e at e at y + ( e at) y = b e at, where we used the main property of the integrating factor, a e at = ( e at). Now the product rule for derivatives implies that the left-hand side above is a total derivative, ( e at y ) = b e at. ( The right-hand above can also be rewritten as a derivative, b e at = b a e at), hence (e at y + b a e at) [( = y + b ) e at] =. a

11 6 G. NAGY ODE september 4, 25 We have succeeded in writing the whole differential equation as a total derivative. differential equation is the total derivative of a potential function, ( ψ(t, y) = y + b ) e at. a The The differential equation for y is a total derivative, dψ (t, y(t)) =, dt so it is simple to integrate, ψ(t, y(t)) = c ( y(t) + b ) e at = c a y(t) = c e at b a. This establishes the Theorem. We solve the example below following the proof of Theorem..2. In this way we see how the ideas in the proof of the Theorem work in a particular example. Example..4: Find all solutions to the constant coefficient equation Solution: Write down the equation in (..8) as follows, y = 2y + 3 (..8) y 2y = 3. Multiply this equation by the integrating factor µ(t) = e 2t, e 2t y 2 e 2t y = 3 e 2t e 2t y + ( e 2t) y = 3 e 2t. The equation on the far right above is (e 2t y) = 3 e 2t. y Rewrite the right hand side above, ( (e 2t y) = 3 2 e 2t). c > Moving terms and reordering factors we get [( y + 3 ) e 2t] =. 2 The equation is a total derivative, ψ =, of the potential function ψ(t, y) = ( y + 3 ) e 2t c = t Now the equation is easy to integrate, ( y + 3 ) e 2t = c. 2 So we get the solutions y(t) = c e 2t 3 2, c R. c < Figure. A few solutions to Eq. (..8) for different c. We now solve the same problem above, but now using the formulas in Theorem..2. Example..5: Find all solutions to the constant coefficient equation y = 2y + 3 (..9)

12 G. NAGY ODE September 4, 25 7 Solution: The equation above is the case of a = 2 and b = 3 in Eq. (..3). Therefore, using these values in the expression for the solution given in Eq. (..4) we obtain y(t) = c e 2t The Initial Value Problem. Sometimes in physics one is not interested in all solutions to a differential equation, but only in those solutions satisfying extra conditions. For example, in the case of Newton s second law of motion for a point particle, one could be interested only in solutions such that the particle is at a particular position at a particular time. Such condition is called an initial condition, and it selects a subset of solutions of the differential equation. An initial value problem means to find a solution to both a differential equation and an initial condition. Definition..3. The initial value problem (IVP) for a constant coefficients first order linear ODE is the following: Given constants a, b, t, y, find a function y solution of y = a y + b, y(t ) = y. (..) The second equation in (..) is called the initial condition of the problem. Although the differential equation in (..) has infinitely many solutions, the associated initial value problem has a unique solution. Theorem..4 (Constant Coefficients IVP). The initial value problem in (..), for given constants a, b, t, y R, and a, has the unique solution ( y(t) = y + b ) e a(t t) b a a. (..) Remark: In case t = the initial condition is y() = y and the solution is ( y(t) = y + b ) e at b a a. The proof of Theorem..4 is just to write the general solution of the differential equation given in Theorem..2, and fix the integration constant c with the initial condition. Proof of Theorem..4: The general solution of the differential equation in (..) is given in Eq. (..4) for any choice of the integration constant c, y(t) = c e at b a. The initial condition determines the value of the constant c, as follows y = y(t ) = c e at b ( c = y + b ) e at. a a Introduce this expression for the constant c into the differential equation in Eq. (..), ( y(t) = y + b ) e a(t t) b a a. This establishes the Theorem. Example..6: Find the unique solution of the initial value problem y = 2y + 3, y() =. (..2) Solution: All solutions of the differential equation are given by y(t) = ce 2t 3 2,

13 8 G. NAGY ODE september 4, 25 where c is an arbitrary constant. The initial condition in Eq. (..2) determines c, = y() = c 3 2 c = 5 2. Then, the unique solution to the initial value problem above is y(t) = 5 2 e2t 3 2. Example..7: Find the solution y to the initial value problem y = 3y +, y() =. Solution: Write the differential equation as y + 3 y =, and multiply the equation by the integrating factor µ = e 3t, which will convert the left-hand side above into a total derivative, e 3t y + 3 e 3t y = e 3t e 3t y + ( e 3t) y = e 3t. This is the key idea, because the derivative of a product implies ( e 3t y ) = e 3t. The exponential e 3t is an integrating factor. Integrate on both sides of the equation, e 3t y = 3 e3t + c. So every solution of the differential equation above is given by y(t) = c e 3t + 3, c R. The initial condition y() = 2 selects only one solution, = y() = c + 3 c = 2 3. We get the solution y(t) = 2 3 e 3t + 3. Notes. This section corresponds to Boyce-DiPrima [3] Section 2., where both constant and variable coefficient equations are studied. Zill and Wright give a more concise exposition in [7] Section 2.3, and a one page description is given by Simmons in [] in Section 2..

14 G. NAGY ODE September 4, Exercises....- Find constants a, b, so that y(t) = (t + 3) e 2t is solution of the IVP y = ay + e 2t, y() = b Follow the steps below to find all solutions of y = 4y + 2 (a) Find the integrating factor µ. (b) Write the equations as a total derivative of a function ψ, that is y = 4y + 2 ψ =. (c) Integrate the equation for ψ. (d) Compute y using part (c) Find all solutions of y = 2y Find the solution of the IVP y = 4y + 2, y() = Find the solution of the IVP dy (t) = 3 y(t) 2, y() =. dt..6.- Express the differential equation y = 6 y + (..3) as a total derivative of a potential function ψ(t, y), that is, find ψ satisfying y = 6 y + ψ =. Integrate the equation for the potential function ψ to find all solutions y of Eq. (..3) Find the solution of the IVP y = 6 y +, y() =.

15 G. NAGY ODE september 4, Linear Variable Coefficient Equations Once we know how to solve linear differential equations with constant coefficients, variable coefficient equations are not so hard to solve. The integrating factor method has is simple to generalize from constant coefficient to variable coefficient equations. We obtain an explicit solution formula for solutions of variable coefficient equations, which generalizes Eq. (..4) in Theorem..2. Initial value problems for variable coefficient equations have unique solutions, just like initial value problems for constant coefficient equations have unique solutions, shown in Theorem..4. Once we know how to solve general linear equations we turn our attention to a particular nonlinear differential equation, the Bernoulli equation. This nonlinear equation has a particular property: It can be transformed into a linear equation by an appropriate change in the unknown function. One then solves the linear equation for the changed function using the integrating factor method. The last step is to transform back the changed function into the original function..2.. Linear Equations with Variable Coefficients. We start this section generalizing Theorem..2 from constant coefficient equations to variable coefficients equations. Theorem.2. (Variable Coefficients). If the functions a, b are continuous, then y = a(t) y + b(t), (.2.) has infinitely many solutions and every solution, y, can be labeled by c R as follows y(t) = c e A(t) + e A(t) e A(t) b(t) dt, (.2.2) where we introduced the function A(t) = a(t) dt, any primitive of the function a. Remark: In the particular case of constant coefficient equations, a primitive (also called antiderivative) for the constant function a is A(t) = at, so the second term in Eq. (.2.2) is e A(t) e A(t) b(t) dt = e at b e at dt = e at( b a e at) = b a. Hence Eq. (.2.2) contains the expression y(t) = c e at b given in Eq. (..4). a Proof of Theorem.2.: Write down the differential equation with y on one side only, y a y = b, and then multiply the differential equation by a function µ, called an integrating factor, The critical step is to choose a function µ such that µ y a µ y = µ b. (.2.3) a(t) µ(t) = µ (t). (.2.4) For any function µ solution of Eq. (.2.4), the differential equation in (.2.3) has the form µ y + µ y = µ b. But the left-hand side is a total derivative of a product of two functions, ( µ y ) = µ b. (.2.5) This is the property we want in an integrating factor, µ. We want to find a function µ such that the left-hand side of the differential equation for y can be written as a total derivative,

16 G. NAGY ODE September 4, 25 just as in Eq. (.2.5). We only need to find one of such functions µ. So we go back to Eq. (.2.4), the differential equation for µ, which is simple to solve, µ = a µ µ µ = a ( ln(µ) ) = a ln(µ) = A + c, where A is a primitive, or antiderivative, of function a, that is A = a, and c is an arbitrary constant. Computing the exponential of both sides we get µ = e A e c µ = c e A, c = e c. Since c is a constant which will cancel out from Eq. (.2.3) anyway, we choose the integration constant c =, hence c =. The integrating function is then µ(t) = e A. This function is an integrating factor, because if we start again at Eq. (.2.3), we get e A y a e A y = e A b e A y + ( e A) y = e A b, where we used the main property of the integrating factor, a e A = ( e A). Now the product rule for derivatives implies that the left-hand side above is a total derivative, ( e A y ) = e A b. The right-hand above can also be rewritten as a derivative of the function K(t) = e A(t) b(t) dt, since K = e A b. We cannot write the function K explicitly, because we do not know the functions A and b. But given the functions A and b we can always compute K. Therefore, the differential equation for y has the form ( e A y K) =. We have succeeded in writing the whole differential equation as a total derivative. differential equation is the total derivative of a potential function, ψ(t, y(t)) = ( e A(t) y(t) K(t) ). The differential equation for y is a total derivative, so it is simple to integrate, dψ (t, y(t)) =, dt ψ(t, y(t)) = c ( e A y K ) = c y = c e A + e A K. If we write the explicit forn of K we get y(t) = c e A(t) + e A(t) This establishes the Theorem. e A(t) b(t) dt. Example.2.: Find all solutions y to the differential equation The y = 3 t y + t5. Solution: Rewrite the equation with y on only one side, y 3 t y = t5.

17 2 G. NAGY ODE september 4, 25 Multiply the differential equation by a function µ, which we determine later, µ(t) (y 3 ) t y = t 5 µ(t) µ(t) y 3 t µ(t) y = t5 µ(t). We need to choose a function µ having the following property, 3 t µ(t) = µ (t) 3 t = µ (t) µ(t) 3 t = ( ln(µ) ) Integrating, 3 ln(µ) = t dt = 3 ln(t) = ln(t 3 ) µ = e ln(t 3 ) µ = t 3. This function µ is an integrating factor of the differential equation. We now go back to the differential equation for y and we multiply it by µ = t 3, t 3( y 3 t y ) = t 3 t 5 t 3 y 3 t 4 y = t 2. ( t Using that 3 t 4 = (t 3 ) and t 2 3 ), = we get 3 ( t t 3 y + (t 3 ) 3 ) y = ( t 3 y ) ( t 3 ) = 3 3 ( t 3 y t3 3 ) =. This last equation is a total derivative of a potential function ψ(t, y) = t 3 y t3. Since the 3 equation is a total derivative, this confirms that we got a correct integrating factor. Now we need to integrate the total derivative, which is simple to do, t 3 y t3 3 = c t 3 y = c + t3 3 where c is an arbitrary constant. y(t) = c t 3 + t6 3,.2.2. The Initial Value Problem. We now generalize Theorems..4 from constant coefficients to variable coefficients equations. But first, let us introduce the initial value problem for a variable coefficients equation, a simple generalization of Def...3. Definition.2.2. The initial value problem (IVP) for a first order linear ODE is the following: Given functions a, b and constants t, y, find a function y solution of y = a(t) y + b(t), y(t ) = y. (.2.6) The second equation in (.2.6) is called the initial condition of the problem. We saw in Theorem.2. that the differential equation in (.2.6) has infinitely many solutions, parametrized by a real constant c. The associated initial value problem has a unique solution though, because the initial condition fixes the constant c. Theorem.2.3 (Variable coefficients IVP). Given continuous functions a, b, with domain (t, t 2 ), and constants t (t, t 2 ) and y R, the initial value problem has the unique solution y on the domain (t, t 2 ), given by where the function A(t) = y = a(t) y + b(t), y(t ) = y, (.2.7) y(t) = y e A(t) + e A(t) t t t e A(s) b(s) ds, (.2.8) t a(s) ds is a particular primitive of function a.

18 G. NAGY ODE September 4, 25 3 Remark: In the particular case of a constant coefficient equation, that is, a, b R, the solution given in Eq. (.2.8) reduces to the one given in Eq. (..). Indeed, A(t) = t t a ds = a(t t ), t t e a(s t ) b ds = b a e a(t t ) + b a. Therefore, the solution y can be written as ( y(t) = y e a(t t) + e a(t t ) b a e a(t t ) + b ) ( = y + b ) e a(t t) b a a a. Proof Theorem.2.3: We follow closely the proof of Theorem..2. From Theorem.2. we know that all solutions to the differential equation in (.2.7) are given by y(t) = c e A(t) + e A(t) e A(t) b(t) dt, for every c R. Let us use again the notation K(t) = e A(t) b(t) dt, and then introduce the initial condition in (.2.7), which fixes the constant c, So we get the constant c, y = y(t ) = c e A(t ) + e A(t ) K(t ). c = y e A(t ) K(t ). Using this expression in the general solution above, ( ) y(t) = y e A(t) K(t ) e A(t) + e A(t) K(t) = y e A(t) A(t) + e A(t)( K(t) K(t ) ). Let us introduce the particular primitives Â(t) = A(t) A(t ) and ˆK(t) = K(t) K(t ), which vanish at t, that is, Â(t) = t t a(s) ds, ˆK(t) = Then the solution y of the IVP has the form which is equivalent to so we conclude that y(t) = y eâ(t) + e A(t) t y(t) = y eâ(t) + e A(t) A(t ) t y(t) = y eâ(t) + eâ(t) t t t e A(s) b(s) ds. t e A(s) b(s) ds t e [A(s) A(t )] b(s) ds, t e Â(s) b(s) ds. Renaming the particular primitive Â simply by A, we then establish the Theorem. We solve the next Example following the main steps in the proof of Theorem.2.3 above. Example.2.2: Find the function y solution of the initial value problem ty + 2y = 4t 2, t >, y() = 2. Solution: We first write the equation above in a way it is simple to see the functions a and b in Theorem.2.3. In this case we obtain y = 2 t y + 4t a(t) = 2, b(t) = 4t. (.2.9) t

19 4 G. NAGY ODE september 4, 25 Now rewrite the equation as and multiply it by a function µ, y + 2 t y = 4t, µ y + 2 t µ y = µ 4t. The function µ is an integrating factor if satisfies 2 t µ = µ ln(µ) = 2 t Multiplying the differential equation for y by µ = t 2 we get ln(µ) = 2 ln(t) = ln(t 2 ) µ = t 2. t 2 y + 2t y = 4t t 2 (t 2 y) = 4t 3. If we write the right hand side also as a derivative, ( t 2 y ) = ( t 4 ) ( t 2 y t 4) =. So the potential function is ψ(t, y(t)) = t 2 y(t) t 4. Integrating on both sides we obtain The initial condition implies that t 2 y t 4 = c t 2 y = c + t 4 y(t) = c t 2 + t2. 2 = y() = c + c = y(t) = t 2 + t2. Example.2.3: Find the solution of the problem given in Example.2.2 using the results of Theorem.2.3. Solution: We find the solution simply by using Eq. (.2.8). First, find the integrating factor function µ as follows: A(t) = t 2 s ds = 2[ ln(t) ln() ] = 2 ln(t) A(t) = ln(t 2 ). The integrating factor is µ(t) = e A(t), that is, µ(t) = e ln(t 2) = e ln(t2 ) µ(t) = t 2. Then, we compute the solution as follows: y(t) = t 2 [2 + 2 ] s 2 4s ds = 2 t 2 + t t 2 4s 3 ds = 2 t 2 + t 2 (t4 ) = 2 t 2 + t2 t 2 y(t) = t 2 + t2.

20 G. NAGY ODE September 4, The Bernoulli Equation. In 696 Jacob Bernoulli struggled for months trying to solve a particular differential equation, now known as Bernoulli s differential equation. He could not solve it, so he organized a contest among his peers to solve the equation. In short time his brother Johann Bernoulli solved it. This was bad news for Jacob because the relation between the brothers was not the best at that time. Later on the equation was solved by Leibniz using a different method than Johann. Leibniz transformed the original nonlinear equation into a linear equation. We now explain Leibniz s idea in more detail. Definition.2.4. A Bernoulli equation in the unknown function y, determined by the functions p, q, and a number n R, is the differential equation y = p(t) y + q(t) y n. (.2.) In the case that n = or n = the Bernoulli equation reduces to a linear equation. The interesting cases are when the Bernoulli equation is nonlinear. We now show in an Example the main idea to solve a Bernoulli equation: To transform the original nonlinear equation into a linear equation. Example.2.4: Find every solution of the differential equation y = y + 2y 5. Solution: This is a Bernoulli equation for n = 5. Divide the equation by the nonlinear factor y 5, y y 5 = y Introduce the function v = /y 4 and its derivative v = 4(y /y 5 ), into the differential equation above, v 4 = v + 2 v = 4 v 8 v + 4 v = 8. The last equation is a linear differential equation for the function v. This equation can be solved using the integrating factor method. Multiply the equation by µ(t) = e 4t, then ( e 4t v ) = 8 e 4t e 4t v = 8 4 e4t + c. We obtain that v = c e 4t 2. Since v = /y 4, y 4 = c e 4t 2 y(t) = ± ( c e 4t 2 ). /4 We now summarize the first part of the calculation in the Example above. Theorem.2.5 (Bernoulli). The function y is a solution of the Bernoulli equation y = p(t) y + q(t) y n, n, iff the function v = /y (n ) is solution of the linear differential equation v = (n )p(t) v (n )q(t).

21 6 G. NAGY ODE september 4, 25 Remark: This result summarizes Laplace s idea to solve the Bernoulli equation. To transform the Bernoulli equation for y, which is nonlinear, into a linear equation for v = /y (n ). One then solves the linear equation for v using the integrating factor method. The last step is to transform back to y = (/v) /(n ). Proof of Theorem.2.5: Divide the Bernoulli equation by y n, y y n = p(t) + q(t). yn Introduce the new unknown v = y (n ) and compute its derivative, v = [ y (n )] = (n )y n y v (t) (n ) = y (t) y n (t). If we substitute v and this last equation into the Bernoulli equation we get v (n ) = p(t) v + q(t) v = (n )p(t) v (n )q(t). This establishes the Theorem. Example.2.5: Given any constants a, b, find every solution of the differential equation y = a y + b y 3. Solution: This is a Bernoulli equation. Divide the equation by y 3, y y 3 = a y 2 + b. Introduce the function v = /y 2 and its derivative v = 2(y /y 3 ), into the differential equation above, v 2 = a v + b v = 2a v 2b v + 2a v = 2b. The last equation is a linear differential equation for v. This equation can be solved using the integrating factor method. Multiply the equation by µ(t) = e 2a t, ( e 2a t v ) = 2b e 2a t e 2a t v = b a e 2a t + c We obtain that v = c e 2a t b a. Since v = /y 2, y 2 = c e 2a t b y(t) = ± a ( c e 2a t b ) /2. a Example.2.6: Find every solution of the equation t y = 3y + t 5 y /3. Solution: Rewrite the differential equation as y = 3 t y + t4 y /3. This is a Bernoulli equation for n = /3. Divide the equation by y /3, y y /3 = 3 t y2/3 + t 4.

22 G. NAGY ODE September 4, 25 7 Define the new unknown function v = /y (n ), that is, v = y 2/3, compute is derivative, v = 2 y, and introduce them in the differential equation, 3 y/3 3 2 v = 3 t v + t4 v 2 t v = 2 3 t4. This is a linear equation for v. Integrate this equation using the integrating factor method. To compute the integrating factor we need to find 2 A(t) = t dt = 2 ln(t) = ln(t2 ). Then, the integrating factor is µ(t) = e A(t). In this case we get µ(t) = e ln(t2) = e ln(t 2 ) µ(t) = t 2. Therefore, the equation for v can be written as a total derivative, ( v t 2 2 t v) = 2 ( v 3 t2 t t3) =. The potential function is ψ(t, v) = v/t 2 (2/9)t 3 and the solution of the differential equation is ψ(t, v(t)) = c, that is, v t 2 2 ( 9 t3 = c v(t) = t 2 c t3) v(t) = c t t5. Once v is known we compute the original unknown y = ±v 3/2, where the double sign is related to taking the square root. We finally obtain ( y(t) = ± c t t5) 3/2. Notes. This section corresponds to Boyce-DiPrima [3] Section 2., and Simmons [] Section 2.. The Bernoulli equation is solved in the exercises of section 2.4 in Boyce- Diprima, and in the exercises of section 2. in Simmons.

23 8 G. NAGY ODE september 4, Exercises Find the general solution of y = y + e 2t Find the solution y to the IVP y = y + 2te 2t, y() = Find the solution y to the IVP t y + 2 y = sin(t) ( π ), y = 2 t 2 π, for t > Find all solutions y to the ODE y (t 2 + )y = 4t Find all solutions y to the ODE ty + n y = t 2, with n a positive integer Find the solutions to the IVP 2ty y =, y() = Find all solutions of the equation y = y 2 sin(t) Find the solution to the initial value problem t y = 2 y + 4t 3 cos(4t), y ( π ) = Find all solutions of the equation y + t y = t y Find all solutions of the equation y = x y + 6x y Find all solutions of the IVP y = y + 3, y() =. y2

24 G. NAGY ODE September 4, Separable Equations.3.. Separable Equations. Often nonlinear differential equations are more complicated to solve than linear equations. Separable equations are an exception they can be solved just by integrating on both sides of the differential equation. We tried this idea to solve linear equations, but it did not work for them. It works for separable equations. Definition.3.. A separable differential equation for the unknown y has the form where h, g are given scalar functions. h(y) y = g(t), If we write the differential equation as y (t) = f(t, y(t)), then the equation is separable iff Example.3.: (a) The differential equation is separable, since it is equivalent to (b) The differential equation f(t, y) = g(t) h(y). y = ( y 2 ) y = t 2 is separable, since it is equivalent to y 2 y = cos(2t) t2 y 2 y + y 2 cos(2t) = { g(t) = t 2, h(y) = y 2. g(t) = cos(2t), h(y) = y 2. The functions g and h are not uniquely defined; another choice in this example is: g(t) = cos(2t), h(y) = y 2. (c) The linear differential equation y = a(t) y is separable, since it is equivalent to g(t) = a(t), y y = a(t) h(y) = y. (d) The equation y = e y + cos(t) is not separable. (e) The constant coefficient linear differential equation y = a y + b is separable, since it is equivalent to (a y + b ) y = g(t) =, h(y) = (a y + b ). (f) The linear equation y = a(t) y + b(t), with a and b nonconstant, is not separable.

25 2 G. NAGY ODE september 4, 25 From the last two examples above we see that linear differential equations, with a, are separable for b constant, and not separable otherwise. Separable differential equations are simple to solve. We simply integrate on both sides of the equation. Theorem.3.2 (Separable Equations). If h, g are continuous, with h, then has infinitely many solutions y satisfying the algebraic equation h(y) y = g(t) (.3.) H(y(t)) = G(t) + c, (.3.2) where c R is arbitrary, H is a primitive (antiderivative) of h, and G is a primitive of g. Remark: The function H is a primitive of function h means H = h, where H = dh/dy. Analogously, G is a primitive of g means G = g, with G = dg/dt. Before we prove this Theorem we solve a particular example. The example will help us identify the functions h, g, H and G, and it will also show how to prove the theorem. Example.3.2: Find all solutions y to the differential equation y (t) = t 2 y 2 (t). (.3.3) Solution: We write the differential equation in (.3.3) in the form h(y) y = g(t), [ y 2 (t) ] y (t) = t 2. In this example the functions h and g defined in Theorem.3.2 are given by h(y) = ( y 2 ), g(t) = t 2. We now integrate with respect to t on both sides of the differential equation, [ y 2 (t) ] y (t) dt = t 2 dt + c, where c is any constant. The integral on the right-hand side can be computed explicitly. The integral on the left-hand side can be done by substitution. The substitution is u = y(t), du = y (t) dt. This substitution on the left-hand side integral above gives, ( u 2 ) du = t 2 dt + c u u3 3 = t3 3 + c. Substitute back the original unknown y into the last expression above and we obtain y(t) y3 (t) = t c. We have solved the differential equation, since there are no derivatives in this last equation. When the solution is given in terms of an algebraic equation, we say that the solution y is given in implicit form. Remark: A primitive of function h(y) = y 2 is function H(y) = y y 3 /3. A primitive of function g(t) = t 2 is function G(t) = t 3 /3. The implicit form of the solution found in Example.3.2 can be written in terms of H and G as follows, y(t) y3 (t) = t3 + c. H(y) = G(t) + c. 3 3 The expression above using H and G is the one we use in Theorem.3.2.

26 G. NAGY ODE September 4, 25 2 Definition.3.3. A solution y of a separable equation h(y) y = g(t) is given in implicit form iff the function y is solution of the algebraic equation H ( y(t) ) = G(t) + c, where H and G are any primitives of h and g. In the case that function H is invertible, the solution y above is given in explicit form iff is written as y(t) = H ( G(t) + c ). In the case that H is not invertible or H is difficult to compute, we leave the solution y in implicit form. Now we show a proof of Theorem.3.2 that is based in an integration by substitution, just like we did in the Example.3.2. Proof of Theorem.3.2: Integrate with respect to t on both sides in Eq. (.3.), h(y(t)) y (t) = g(t) h(y(t)) y (t) dt = g(t) dt + c, where c is an arbitrary constant. Introduce on the left-hand side of the second equation above the substitution u = y(t), du = y (t) dt. The result of the substitution is h(y(t)) y (t) dt = h(u)du h(u) du = g(t) dt + c. To integrate on each side of this equation means to find a function H, primitive of h, and a function G, primitive of g. Using this notation we write H(u) = h(u) du, G(t) = g(t) dt. Then the equation above can be written as follows, H(u) = G(t) + c. Substitute u back by y(t). We arrive to the algebraic equation for the function y, H ( y(t) ) = G(t) + c. This establishes the Theorem. In the Example below we solve the same problem than in Example.3.2 but now we just use the result of Theorem.3.2. Example.3.3: Use the formula in Theorem.3.2 to find all solutions y to the equation y (t) = t 2 y 2 (t). (.3.4) Solution: Theorem.3.2 tell us how to obtain the solution y. Writing Eq. (.3.3) as ( y 2 ) y (t) = t 2, we see that the functions h, g are given by h(u) = u 2, g(t) = t 2. Their primitive functions, H and G, respectively, are simple to compute, h(u) = u 2 H(u) = u u3 3, g(t) = t 2 G(t) = t3 3.

27 22 G. NAGY ODE september 4, 25 Then, Theorem.3.2 implies that the solution y satisfies the algebraic equation where c R is arbitrary. y(t) y3 (t) 3 = t3 3 + c, (.3.5) Remark: Sometimes it is simpler to remember ideas than formulas. So one can solve a separable equation as we did in Example.3.2, instead of using the solution formulas, as in Example.3.3. (Although in the case of separable equations both methods are very close.) In the next Example we show that an initial value problem can be solved even when the solutions of the differential equation are given in implicit form. Example.3.4: Find the solution of the initial value problem y (t) = t 2 y 2, y() =. (.3.6) (t) Solution: From Example.3.2 we know that all solutions to the differential equation in (.3.6) are given by y(t) y3 (t) = t c, where c R is arbitrary. This constant c is now fixed with the initial condition in Eq. (.3.6) y() y3 () = c 3 = c c = 2 y(t) y3 (t) = t So we can rewrite the algebraic equation defining the solution functions y as the roots of a cubic polynomial, y 3 (t) 3y(t) + t =. We present now a few more Examples. Example.3.5: Find the solution of the initial value problem y (t) + y 2 (t) cos(2t) =, y() =. (.3.7) Solution: The differential equation above is separable, with therefore, it can be integrated as follows: Again the substitution y (t) y 2 (t) = cos(2t) g(t) = cos(2t), h(y) = y 2, u = y(t), y (t) y 2 (t) dt = du = y (t) dt cos(2t) dt + c. implies that du u 2 = cos(2t) dt + c u = sin(2t) + c. 2 Substitute the unknown function y back in the equation above, y(t) = sin(2t) + c. 2

28 G. NAGY ODE September 4, The solution is given in implicit form. However, in this case is simple to solve this algebraic equation for y and we obtain the following explicit form for the solutions, The initial condition implies that y(t) = 2 sin(2t) 2c. = y() = 2 c =. 2c So, the solution to the IVP is given in explicit form by y(t) = 2 sin(2t) + 2. Example.3.6: Follow the proof in Theorem.3.2 to find all solutions y of the ODE y (t) = 4t t3 4 + y 3 (t). Solution: The differential equation above is separable, with therefore, it can be integrated as follows: [ 4 + y 3 (t) ] y (t) = 4t t 3 g(t) = 4t t 3, h(y) = 4 + y 3, [4 + y 3 (t) ] y (t) dt = (4t t 3 ) dt + c. Again the substitution u = y(t), du = y (t) dt implies that (4 + u 3 ) du = (4t t 3 ) dt + c. 4u + u4 4 = 2t2 t4 4 + c. Substitute the unknown function y back in the equation above and calling c = 4c we obtain the following implicit form for the solution, y 4 (t) + 6y(t) 8t 2 + t 4 = c. Example.3.7: Find the explicit form of the solution to the initial value problem y (t) = 2 t + y(t) Solution: The differential equation above is separable with Their primitives are respectively given by, g(t) = 2 t, h(u) = + u. y() =. (.3.8) g(t) = 2 t G(t) = 2t t2 u2, h(u) = + u H(u) = u Therefore, the implicit form of all solutions y to the ODE above are given by y(t) + y2 (t) 2 = 2t t2 2 + c,

29 24 G. NAGY ODE september 4, 25 with c R. The initial condition in Eq. (.3.8) fixes the value of constant c, as follows, y() + y2 () = + c = c c = 3 2. We conclude that the implicit form of the solution y is given by y(t) + y2 (t) = 2t t , y2 (t) + 2y(t) + (t 2 4t 3) =. The explicit form of the solution can be obtained realizing that y(t) is a root in the quadratic polynomial above. The two roots of that polynomial are given by y ± (t) = 2[ 2 ± 4 4(t2 4t 3) ] y ± (t) = ± t 2 + 4t + 4. We have obtained two functions y + and y. However, we know that there is only one solution to the IVP. We can decide which one is the solution by evaluating them at the value t = given in the initial condition. We obtain y + () = + 4 =, y () = 4 = 3. Therefore, the solution is y +, that is, the explicit form of the solution is y(t) = + t 2 + 4t Euler Homogeneous Equations. Sometimes a differential equation is not separable but it can be transformed into a separable equation changing the unknown function. This is the case for differential equations of the form N(t, y) y (t) + M(t, y) =, where the functions M, N are both homogeneous of the same degree. Definition.3.4. A function f, on the variables t, y, is homogeneous of degree n iff for every t, y on the domain of f and for all c R holds f(ct, cy) = c n f(t, y). An example of an homogeneous function is the energy of a thermodynamical system, such as a gas in a bottle. The energy, E, of a fixed amount of gas is a function of the gas entropy, S, and the gas volume, V. Such energy is an homogeneous function of degree one, E(cS, cv ) = c E(S, V ), for all c R. Example.3.8: Show that the functions f and f 2 are homogeneous and find their degree, f (t, y) = t 4 y 2 + ty 5 + t 3 y 3, f 2 (t, y) = t 2 y 2 + ty 3. Solution: The function f is homogeneous of degree 6, since f (ct, cy) = c 4 t 4 c 2 y 2 + ct c 5 y 5 + c 3 t 3 c 3 y 3 = c 6 (t 4 y 2 + ty 5 + t 3 y 3 ) = c 6 f(t, y). Notice that the sum of the powers of t and y on every term is 6. Analogously, function f 2 is homogeneous degree 4, since f 2 (ct, cy) = c 2 t 2 c 2 y 2 + ct c 3 y 3 = c 4 (t 2 y 2 + ty 3 ) = c 2 f 2 (t, y). Again, the sum of the powers of t and y on every term is 4.

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