A Second Course in Elementary Ordinary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 26 Calculus of Matrix-Valued Functions of a Real Variable 4 27 nth Order Linear Differential Equations:Existence and Unique- ness 18 28 The General Solution of nth Order Linear Homogeneous Equa- tions 25 29 Fundamental Sets and Linear Independence 34 30 Higher Order Homogeneous Linear Equations with Constant Coefficients 40 31 Non Homogeneous nth Order Linear Differential Equations 47 32 Existence and Uniqueness of Solution to Initial Value First Order Linear Systems 56 33 Homogeneous First Order Linear Systems 63 34 First Order Linear Systems: Fundamental Sets and Linear Independence 74 35 Homogeneous Systems with Constant Coefficients 83 36 Homogeneous Systems with Constant Coefficients: Complex Eigenvalues 93 37 Homogeneous Systems with Constant Coefficients: Repeated Eigenvalues 97 38 Nonhomogeneous First Order Linear Systems 108 39 Solving First Order Linear Systems with Diagonalizable Con- stant Coefficients Matrix 118 40 Solving First Order Linear Systems Using Exponential Ma- trix 126 2 41 The Laplace Transform: Basic Definitions and Results 131 42 Further Studies of Laplace Transform 142 43 The Laplace Transform and the Method of Partial Fractions155 44 Laplace Transforms of Periodic Functions 162 45 Solving Systems of Differential Equations Using Laplace Trans- form 171 46 Convolution Integrals 178 47 The Dirac Delta Function and Impulse Response 186 48 Numerical Solutions to ODEs: Euler’s Method and its Vari- ants 194 3 26 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 a11 a12 ... a1n a21 a22 ... a2n A = ... ... ... ... 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. The zero matrix 0 is the matrix whose entries are all 0. The n × n identity matrix In is a square matrix whose main diagonal consists of 10s and the off diagonal entries are all 0. A matrix A can be represented with the following compact notation A = (aij). The entry aij is located in the ith row and jth column. Example 26.1 Consider the matrix −5 0 1 A(t) = 10 −2 0 −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 4 An m × n array whose entries are functions of a real variable defined on a common interval is called a matrix function. Thus, the matrix a11(t) a12(t) a13(t) A(t) = a21(t) a22(t) a23(t) a31(t) a32(t) a33(t) is a 3 × 3 matrix function whereas the matrix x1(t) x(t) = x2(t) 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. 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 26.2 Solve the following matrix equation for a, b, c, and d a − b b + c 8 1 = 3d + c 2a − 4d 7 6 Solution. Equating corresponding entries we get the system a − b = 8 b + c = 1 c + 3d = 7 2a − 4d = 6 Adding the first two equations to obtain a + c = 9. Adding 4 times the third equation to 3 times the last equation to obtain 6a + 4c = 46 or 3a + 2c = 23. 5 Solving the two equations in a and c one finds a = 5 and c = 4. Hence, b = −3 and d = 1 (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 ele- ments (A + B)(t) = A(t) + B(t) = (aij(t) + bij(t)) 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)) (iv) Scalar Multiplication: If α is a real number and A(t) = (aij(t)) is an m × n matrix then (αA)(t) is the m × n matrix obtained by multiplying the entries of A by the number α; that is, (αA)(t) = αA(t) = (αaij(t)) (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 26.3 1 2 2 −1 A = , B = 3 2 −3 4 Show that AB 6= BA. Hence, matrix multiplication is not commutative. 6 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 if and only if there is an n × n matrix B(t) such that AB(t) = BA(t) = I where I is the matrix whose main diagonal consists of the number 1 and 0 elsewhere. We denote the inverse of A(t) by A−1(t). Example 26.4 Find the inverse of the matrix a b A = c d given that ad − bc 6= 0. The quantity ad − bc is called the determinant of A and is denoted by detA Solution. Suppose that x y A−1 = z t Then x y a b 1 0 = z t c d 0 1 This implies that ax + cy bx + dy 1 0 = az + ct bz + dt 0 1 Hence, ax + cy = 1 bx + dy = 0 az + ct = 0 bz + dt = 1 Applying the method of elimination to the first two equations we find 7 d −b x = ad−bc and y = ad−bc Applying the method of elimination to the last two equations we find −c a z = ad−bc and t = ad−bc Hence, 1 d −b A−1 = ad − bc −c a 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 with components x1, x2, ··· , xn by ||x|| = |x1| + |x2| + ··· + |xn|. From this definition one notices the following properties: (i) If ||x|| = 0 then |x1| + |x2| + ··· + |xn| = 0 and this implies that |x1| = |x2| = ··· = |xn| = 0. Hence, x = 0. (ii) If α is a scalar then ||αx|| = |αx1| + |αx2| + ··· + |αxn| = |α|(|x1| + |x2| + ··· + |xn|) = |α|||x||. (iii) If x is vector function with components x1, x2, ··· , xn and y with com- ponents y1, y2, ··· , yn then ||x + y|| = |x1 + y1| + |x2 + y2| + ··· + |xn + yn| ≤ (|x1| + |x2| + ··· + |xn|) + (|y1| + |y2| + ··· + |yn|) = ||x|| + ||y|| Limits of Matrix Functions If A(t) = (aij(t)) is an m × n matrix such that limt→t0 aij(t) = Lij exists for all 1 ≤ i ≤ m and 1 ≤ j ≤ n then we define lim A(t) = (Lij) t→t0 Example 26.5 Suppose that t2 − 5t t3 A(t) = 2t 3 Find limt→1 A(t). 8 Solution. lim (t2 − 5t) lim t3 −4 1 lim A(t) = t→1 t→1 = t→1 limt→1 2t limt→1 3 2 3 If one or more of the component function limits does not exist, then the limit of the matrix does not exist. For example, if t t−1 A(t) = 0 et 1 then limt→0 A does not exist since limt→0 t does not exist. We say that A(t) is continuous at t = t0 if lim A(t) = A(t0) t→t0 Example 26.6 Show that the matrix t t−1 A(t) = 0 et is continuous at t = 1. Solution. Since 2 1/2 lim A(t) = 2 = A(1) t→1 0 e then A(t) is continuous at t = 1 Most properties of limits for functions of a single variable are also valid for limits of matrix functions. Matrix Differentiation Let A(t) be an m × n matrix such that each entry is a differentiable function of t.