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1 ORDINARY DIFFERENTIAL EQUATIONS . FOR ENGINEERS . THE LECTURE NOTES FOR MATH- 263 (2011). ORDINARY DIFFERENTIAL EQUATIONS . FOR ENGINEERS . JIAN-JUN XU. Department of Mathematics and Statistics, McGill University Kluwer Academic Publishers Boston/Dordrecht/London Contents 1. INTRODUCTION 1. 1 De nitions and Basic Concepts 1. ORDINARY Di erential Equation (ODE) 1. Solution 1. Order n of the DE 2. Linear Equation: 2. Homogeneous Linear Equation: 3. Partial Di erential Equation (PDE) 3. General Solution of a Linear Di erential Equation 3. A System of ODE's 4. 2 The Approaches of Finding Solutions of ODE 5. Analytical Approaches 5. Numerical Approaches 5. 2. FIRST ORDER DIFFERENTIAL EQUATIONS 7. 1 Linear Equation 7. Linear homogeneous equation 8. Linear inhomogeneous equation 8. 2 Nonlinear EQUATIONS (I) 11. separable EQUATIONS . 11. Logistic Equation 14. Fundamental Existence and Uniqueness Theorem 16.

2 Bernoulli Equation: 17. Homogeneous Equation: 18. 3 Nonlinear EQUATIONS (II) Exact Equation and Integrating Factor 20. Exact EQUATIONS . 20. v vi ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS . 4 Integrating Factors. 21. Contents vii 3. N-TH ORDER DIFFERENTIAL EQUATIONS 25. 1 Introduction 25. 2 (*)Fundamental Theorem of Existence and Uniqueness 26. Theorem of Existence and Uniqueness (I) 26. Theorem of Existence and Uniqueness (II) 27. Theorem of Existence and Uniqueness (III) 27. 3 Linear EQUATIONS 27. Basic Concepts and General Properties 27. Linearity 28. Superposition of Solutions 29. ( ) Kernel of Linear operator L(y) 29. New Notations 29. 4 Basic Theory of Linear Di erential EQUATIONS 30. Basics of Linear Vector Space 31. Dimension and Basis of Vector Space, Fundamental Set of Solutions of Eq. 31. Linear Independency 31. Wronskian of n-functions 34. De nition 34.

3 Theorem 1 35. Theorem 2 36. The Solutions of L[y] = 0 as a Linear Vector Space 38. 5 Finding the Solutions in terms of the Method with Di erential Operators 38. Solutions for EQUATIONS with Constants Coe cients 38. Basic Equalities (I). 39. Cases (I) 39. Cases (II) 40. Cases (III) 40. Summary 42. Theorem 1 43. 6 Solutions for EQUATIONS with Variable Coe cients 46. Euler EQUATIONS 47. 7 Finding the Solutions in terms of the Method with Undetermined Parameters 49. Solutions for EQUATIONS with Constants Coe cients 50. Basic Equalities (II) 50. viii ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS . Cases (I) ( r1 > r2 ) 51. Cases (II) ( r1 = r2 ) 51. Cases (III) ( r1,2 = i ) 54. Solutions for Euler EQUATIONS 54. Basic Equalities (III) 54. Cases (I) ( r1 = r2 ) 55. Cases (II) ( r1 = r2 ) 55. Cases (III) ( r1,2 = i ) 56. (*) Exact EQUATIONS 58. 8 Finding a Particular Solution for Inhomogeneous Equation 59.

4 The Annihilator and the Method of Undetermined Constants 59. The Annihilators for Some Types of Functions 60. Basic Properties of the Annihilators 61. The Basic Theorems of the Annihilator Method 62. The Method of Variation of Parameters 67. Reduction of Order 71. 4. LAPLACE TRANSFORMS 75. 1 Introduction 75. 2 Laplace Transform 77. De nition 77. Piecewise Continuous Function 77. Laplace Transform 77. Existence of Laplace Transform 77. 3 Basic Properties and Formulas of Laplace Transform 78. Linearity of Laplace Transform 78. Laplace Transforms for f (t) = eat 78. Laplace Transforms for f (t) = {sin(bt) and cos(bt)}. 78. Laplace Transforms for {eat f (t); f (bt)} 79. Laplace Transforms for {tn f (t)} 79. Laplace Transforms for {f (t)} 80. 4 Inverse Laplace Transform 80. Theorem 1: 80. Theorem 2: 81. De nition 81. Contents ix 5 Solve IVP of DE's with Laplace Transform Method 81.

5 Example 1 81. Example 2 83. Example 3 85. 6 Further Studies of Laplace Transform 86. Step Function 86. De nition 86. Some basic operations with the step function 86. Laplace Transform of Unit Step Function 87. Impulse Function 93. De nition 93. Laplace Transform of Impulse Function 94. Convolution Integral 95. Theorem 95. The properties of convolution integral 95. 5. SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS . 97. 1 Series Solutions near an ORDINARY Point 97. Introduction 97. Series Solutions near an ORDINARY Point 98. 2 Series Solution near a Regular Singular Point 101. Introduction 101. Series Form of Solutions near a Regular Singular Point 101. Case (I): The roots (r1 r2 = N ) 103. Case (II): The roots (r1 = r2 ) 104. Case (III): The roots (r1 r2 = N > 0) 105. Summary 107. 3 (*)Bessel Equation 114. The Case of Non-integer 115. The Case of = m with m an integer 0 116.

6 4 Behaviors of Solutions near the Regular Singular Point x=0 118. Case (I): r1 r2 = N 119. Case (II): r1 = r2 119. Case (III): r1 r2 = N = 0 120. 6. ( ) SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 121. 1 Introduction 121. x ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS . (2 2) System of Linear EQUATIONS 122. Case 1: > 0 122. Case 2: < 0 123. Case 3: = 0 124. 2 Solutions for (n n) Homogeneous Linear System 128. Case (I): (A) is non-defective matrix 128. Case (II): (A) has a pair of complex conjugate eigen-values 130. Case (III): (A) is a defective matrix 130. 3 Solutions for Non-homogeneous Linear System 133. Variation of Parameters Method 133. Chapter 1. INTRODUCTION. 1. De nitions and Basic Concepts ORDINARY Di erential Equation (ODE). An equation involving the derivatives of an unknown function y of a single variable x over an interval x (I). More clearly and precisely speaking, a well de ned ODE must the following features: It can be written in the form: F [x, y, y , y , , y n ] = 0; ( ).

7 Where the mathematical expression on the right hand side contains (1). variable x, (2). function y of x, and (3). some derivatives of y with respect to x;. The values of variables x, y must be speci ed in a certain number eld, such as N , R, or C;. The variation region of variable x of Eq. must be speci ed, such as x (I) = (a, b). Solution Any function y = f (x) which satis es this equation over the interval (I) is called a solution of the ODE. More clearly speaking, function (x). is called a solution of the give EQ. ( ), if the following requirements are satis es: The function (x) is de ned in the region x (I);. The function (x) is di erentiable, hence, { (x), , (n) (x)} all exit, in the region x (I);. 1. 2 ORDINARY DIFFERENTIAL EQUATIONS FOR ENGINEERS . With the replacements of the variables y, y , , y (n) in by the functions (x), (x), , (n) (x), the EQ. ( ) becomes an identity over x (I).

8 In other words, the right hand side of Eq. ( ) becomes to zero for all x (I). For example, one can verify that y = e2x is a solution of the ODE. y = 2y, x ( , ), and y = sin(x2 ) is a solution of the ODE. xy y + 4x3 y = 0, x ( , ). Order n of the DE. An ODE is said to be order n, if y (n) is the highest order derivative occurring in the equation. The simplest rst order ODE is y = g(x). Note that the expression F on the right hand side of an n-th order ODE: F [x, y, y , .. , y (n) ] = 0 can be considered as a function of n + 2. variables (x, u0 , u1 , .. , un ). Namely, one may write F (x, u0 , u1 , , un ) = 0. Thus, the EQUATIONS xy + y = x3 , y + y 2 = 0, y + 2y + y = 0. which are examples of ODE's of second order, rst order and third order respectively, can be in the forms: F (x, u0 , u1 , u2 ) = xu2 + u0 x3 , F (x, u0 , u1 ) = u1 + u20 , F (x, u0 , u1 , u2 , u3 ) = u3 + 2u1 + u0.

9 Respectively. Linear Equation: If the function F is linear in the variables u0 , u1 , .. , un , which means every term in F is proportional to u0 , u1 , .. , un , the ODE is said to be linear. If, in addition, F is homogeneous then the ODE is said to be homogeneous. The rst of the above examples above is linear are linear, the second is non-linear and the third is linear and homogeneous. The general n-th order linear ODE can be written n d y d y n 1 dy an (x) dxn + an 1 (x) dxn 1 + + a1 (x) dx +a0 (x)y = b(x). INTRODUCTION 3. Homogeneous Linear Equation: The linear DE is homogeneous, if and only if b(x) 0. Linear homo- geneous EQUATIONS have the important property that linear combinations of solutions are also solutions. In other words, if y1 , y2 , .. , ym are solu- tions and c1 , c2 , .. , cm are constants then c1 y1 + c2 y2 + + cm ym is also a solution. Partial Di erential Equation (PDE).

10 An equation involving the partial derivatives of a function of more than one variable is called PED. The concepts of linearity and homo- geneity can be extended to PDE's. The general second order linear PDE. in two variables x, y is 2 2 2. a(x, y) xu2 + b(x, y) x y u + c(x, y) yu2 + d(x, y) u x +e(x, y) u y + f (x, y)u = g(x, y). Laplace's equation 2u 2u + 2 =0. x2 y is a linear, homogeneous PDE of order 2. The functions u = log(x2 +y 2 ), u = xy, u = x2 y 2 are examples of solutions of Laplace's equation. We will not study PDE's systematically in this course. General Solution of a Linear Di erential Equation It represents the set of all solutions, , the set of all functions which satisfy the equation in the interval (I). For example, given the di erential equation y = 3x2 . Its general solution is y = x3 + C. where C is an arbitrary constant. To select a speci c solution, one needs to determine the constant C with some additional conditions.