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A Second Course in Ordinary Differential Equations

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Lecture Notes in Mathematics Arkansas Tech University Department of Mathematics A Second Course in Ordinary Differential Equations Marcel B. Finan c All Rights Reserved 2015 Edition To my children Amin and Nadia Preface The present text, while mathematically rigorous, is readily accessible to stu- dents of either mathematics or engineering. In addition to all the theoretical considerations, there are numerous worked examples drawn from engineering and physics. It is important to have a good understanding of the theory underlying it, rather than just a cursory knowledge of its application. This text provides that understanding. Marcel B. Finan Russellville, Arkansas May 2015 i ii PREFACE Contents Prefacei 1 nth Order Linear Ordinary Differential Equations3 1.1 Calculus of Matrix-Valued Functions of a Real Variable....4 1.2 Uniform Convergence and Weierstrass M-Test......... 16 1.3 Existence and Uniqueness Theorem............... 26 1.4 The General Solution of nth Order Linear Homogeneous Equa- tions................................ 38 1.5 Existence of Many Fundamental Sets.............. 47 1.6 nth Order Homogeneous Linear Equations with Constant Co- efficients.............................. 55 1.7 Review of Power Series...................... 66 1.8 Series Solutions of Linear Homogeneous Differential Equations 75 1.9 Non-Homogeneous nth Order Linear Differential Equations.. 80 2 First Order Linear Systems 87 2.1 Existence and Uniqueness of Solution to Initial Value First Order Linear Systems....................... 88 2.2 Homogeneous First Order Linear Systems........... 96 2.3 First Order Linear Systems: Fundamental Sets and Linear Independence........................... 107 2.4 Homogeneous Systems with Constant Coefficients: Distinct Eigenvalues............................ 116 2.5 Homogeneous Systems with Constant Coefficients: Complex Eigenvalues............................ 126 2.6 Homogeneous Systems with Constant Coefficients: Repeated Eigenvalues............................ 131 1 2 CONTENTS Answer Key 141 Index 171 Chapter 1 nth Order Linear Ordinary Differential Equations This chapter is devoted to extending the results for solving second order ordinary differential equations to higher order ODEs. 3 4CHAPTER 1. N TH ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS 1.1 Calculus of Matrix-Valued Functions of a Real Variable In establishing the existence result for second and higher order linear differ- ential equations one transforms the equation into a linear system and tries to solve such a system. This procedure requires the use of concepts such as the derivative of a matrix whose entries are functions of t; the integral of a matrix, and the exponential matrix function. Thus, techniques from matrix theory play an important role in dealing with systems of differential equations. The present section introduces the necessary background in the calculus of matrix functions. Matrix-Valued Functions of a Real Variable A matrix A of dimension m × n is a rectangular array of the form 2 3 a11 a12 ::: a1n 6 a21 a22 ::: a2n 7 A = 6 7 4 ::: ::: ::: ::: 5 am1 am2 ::: amn where the aij's are the entries of the matrix, m is the number of rows, n is the number of columns. When m = n we call the matrix a square matrix. The zero matrix 0 is the matrix whose entries are all 0. The n×n identity 0 matrix In is a square matrix whose main diagonal consists of 1 s and the off diagonal entries are all 0. A matrix A can be represented with the following th th compact notation A = [aij]: The entry aij is located in the i row and j column. Example 1.1.1 Consider the matrix 2−5 0 13 A = 4 10 −2 05 −5 2 −7 Find a22; a32; and a23: Solution. The entry a22 is in the second row and second column so that a22 = −2: Similarly, a32 = 2 and a23 = 0 1.1. CALCULUS OF MATRIX-VALUED FUNCTIONS OF A REAL VARIABLE5 An m × n array whose entries are functions of a real variable defined on a common interval I is called a matrix function. Thus, the matrix 2 3 a11(t) a12(t) a13(t) A(t) = 4 a21(t) a22(t) a23(t) 5 a31(t) a32(t) a33(t) is a 3 × 3 matrix function whereas the matrix 2 3 x1(t) x(t) = 4 x2(t) 5 x3(t) is a 3 × 1 matrix function also known as a vector-valued function. We will denote an m × n matrix function by A(t) = [aij(t)] where aij(t) is the entry in the ith row and jth coloumn. Arithmetic of Matrix Functions All the familiar rules of matrix arithmetic hold for matrix functions as well. (i) Equality: Two m × n matrices A(t) = [aij(t)] and B(t) = [bij(t)] are said to be equal if and only if aij(t) = bij(t) for all 1 ≤ i ≤ m and 1 ≤ j ≤ n and all t 2 I: That is, two matrices are equal if and only if all corresponding elements are equal. Notice that the matrices must be of the same dimension. Example 1.1.2 Solve the following matrix equation. t2 − 3t + 2 1 0 1 = 2 −1 2 t2 − 4t + 2 Solution. Equating corresponding entries we get t2 − 3t + 2 = 0 and t2 − 4t + 2 = −1: Solving these equations, we find t = 1 and t = 2 for the firstr equation and t = 1 and t = 3 for the second equation. Hence, t = 1 is the common value (ii) Addition: If A(t) = [aij(t)] and B(t) = [bij(t)] are m × n matrices then the sum is a new m × n matrix obtained by adding corresponding elements (A + B)(t) = A(t) + B(t) = [aij(t) + bij(t)]: 6CHAPTER 1. N TH ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS Matrices of different dimensions cannot be added. (iii) Subtraction: Let A(t) = [aij(t)] and B(t) = [bij(t)] be two m × n matrices. Then the difference (A − B)(t) is the new matrix obtained by subtracting corresponding elements,that is (A − B)(t) = A(t) − B(t) = [aij(t) − bij(t)]: Example 1.1.3 Consider the following matrices t − 1 t2 t −1 A(t) = ; B(t) = : 2 2t + 1 0 t + 2 Find A(t) + B(t) and A(t) − B(t): Solution. We have t − 1 t2 t −1 2t − 1 t2 − 1 A(t) + B(t) = + = 2 2t + 1 0 t + 2 2 3t + 3 and t − 1 t2 t −1 −1 t2 + 1 A(t) − B(t) = − = 2 2t + 1 0 t + 2 2 t − 1 (iv) Matrix Multiplication by a function: If α(t) is a function and A(t) = [aij(t)] is an m × n matrix then α(t)A)(t) is the m × n matrix obtained by multiplying the entries of A(t) by α(t); that is, α(t)A)(t) = [α(t)aij(t)]: Example 1.1.4 Find t2A(t) where A(t) is the matrix defined in Example 1.1.3. Solution. We have t3 − t2 t4 t2A(t) = 2t2 2t3 + t2 1.1. CALCULUS OF MATRIX-VALUED FUNCTIONS OF A REAL VARIABLE7 (v) Matrix Multiplication: If A(t) is an m×n matrix and B(t) is an n×p matrix then the matrix AB(t) is the m × p matrix AB(t) = [cij(t)] where n X cij(t) = aik(t)bkj(t): k=1 That is, the cij entry is obtained by multiplying componentwise the ith row of A(t) by the jth column of B(t). It is important to realize that the order of the multiplicands is significant, in other words AB(t) is not necessarily equal to BA(t). In mathematical terminology matrix multiplication is not commutative. Example 1.1.5 1 2 2 −1 A = ; B = 3 2 −3 4 Show that AB 6= BA: Hence, matrix multiplication is not commutative. Solution. Using the definition of matrix multiplication we find −4 7 −1 2 AB = ; BA = : 0 5 9 2 Hence, AB 6= BA (vi) Inverse: An n×n matrix A(t) is said to be invertible or non-singular if and only if there is an n × n matrix B(t) such that AB(t) = BA(t) = In: We denote the inverse of A(t) by A−1(t): Example 1.1.6 Find the inverse of the matrix t 1 A = 1 t given that t 6= ±1: 8CHAPTER 1. N TH ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS Solution. Suppose that b (t) b (t) A−1 = 11 12 b21(t) b22(t) Then b (t) b (t) t 1 1 0 11 12 = b21(t) b22(t) 1 t 0 1 This implies that tb (t) + b (t) b (t) + tb (t) 1 0 11 12 11 12 = tb21(t) + b22(t) b21(t) + tb22(t) 0 1 Hence, tb11(t) + b12(t) = 1 b11(t) + tb12(t) = 0 tb21(t) + b22(t) = 0 b21(t) + tb22(t) = 1: Applying the method of elimination to the first two equations we find t −1 b11(t) = t2−1 and b12(t) = t2−1 : Applying the method of elimination to the last two equations we find −1 t b21(t) = t2−1 and b22(t) = t2−1 : Hence, 1 t −1 A−1 = t2 − 1 −1 t Norm of a Vector Function The norm of a vector function will be needed in the coming sections. In one dimension a norm is known as the absolute value. In multidimenesion, we define the norm of a vector function x(t) with components x1(t); x2(t); ··· ; xn(t) by jjx(t)jj = jx1(t)j + jx2(t)j + ··· + jxn(t)j: From this definition one notices the following properties: (i) If jjx(t)jj = 0 then jx1(t)j + jx2(t)j + ··· + jxn(t)j = 0 and this implies that 1.1.
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