9 - Matrix Methods for Linear Systems

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1 9 - Matrix Methods for Linear Systems 9.4 Linear Systems in Normal Form Homework: p # ü Introduction Consider a system of n linear differential equations given by x 1 x 2 x n = a 11 HtL x 1 HtL + a 12 HtL x 2 HtL + + a 1 n HtL x n HtL + f 1 HtL = a 21 HtL x 1 HtL + a 22 HtL x 2 HtL + + a 2 n HtL x n HtL + f 2 HtL = a n 1 HtL x 1 HtL + a n 2 HtL x 2 HtL + + a nn HtL x n HtL + f n HtL We say that this system is in normal form if it can be written as (1) x' HtL = AHtL xhtl + fhtl where xhtl = x 1 HtL x 2 HtL x n HtL, fhtl = f 1 HtL f 2 HtL f n HtL, and AHtL = a 11 HtL a 12 HtL a 1 n HtL a 21 HtL a 22 HtL a 2 n HtL a n 1 HtL a n 2 HtL a nn HtL The system is said to be homogeneous when fhtl 0, otherwise it is said to be nonhomogeneous. When the elements of A are constants, the system is said to have constant coefficients. The vector function xhtl is a solution of the initial value problem for the normal system (1) if it satisfies the system on x 1, 0 an inteval I and also satisfies the initial condition xht 0 L = x 0, where t 0 is a given point of I and x 0 = x 2, 0 x n,0. Theorem 2 - Existence and Uniqueness Suppose AHtL and fhtl are continuous on an open interval I that contains the point t 0. Then, for any choice of the initial vector x 0, there exists a unique solution xhtl on the whole interval I to the initial value problem x' HtL = AHtL xhtl + fhtl, xht 0 L = x 0 If we rewrite the differential equation as x' - Ax= f and define the operator L as L@xD := x' - Ax, then we can express the system (1) in operator form as L@xD = f. Recall that an operator like L is a linear operator, which means for that for any scalars a and b and differenitable vector functions x and y, we have L@a x + b yd = al@xd + bl@yd. Since L is a linear operator, if x 1, x 2,.., x n are solutions to the homogeneous system x' = Ax, then any linear combination c 1 x 1 + c 2 x c n x n is also a solution.

2 2 CH_09_notes.nb Definition 1 - Linear Dependence of Vector Functions The m vector functions x 1, x 2,.., x m are said to be linearly dependent on an interval I if there exist constants c 1, c 2,..., c m not all zero, such that (3) c 1 x 1 + c 2 x c m x m = 0 for all t in I. If the vectors are not linearly dependent, they are said to be linearly independent on I. Definition 2 - Wronskian The Wronskian of n vector functions x 1 HtL = colhx 1, 1, x 1, 2,..., x 1, n L,..., x n HtL = colhx n,1, x n,2,..., x n, n L is defined to be the real-valued function x 1, 1 HtL x 1, 2 HtL x 1, n HtL x 2, 1 HtL x 2, 2 HtL x 2, n HtL W@x 1,..., x n DHtL := x n,1 HtL x n,2 HtL x n, n HtL A set of n solutions x 1, x 2,.., x n of the homogeneous system x' = Ax on some interval I is linearly independent on I if and only if their Wronskian is never zero on I. Theorem 3 - Representation of Solutions (Homogeneous Case) Let x 1, x 2,.., x n be n linearly independent solutions to the homogeneous system (5) x' HtL = A HtL xhtl on the interval I, where A HtL is an nμn matrix function continuous on I. Then every solution to (5) on I can be expressed in the form (6) xhtl = c 1 x 1 HtL + c 2 x 2 HtL + + c n x n HtL where c 1, c 2,..., c n are constants. A set of solutions 8x 1,..., x n < that are linearly independent on I is called a fundamental solution set for (5) on I. The fundamental matrix for (5) is the matrix given by XHtL = x 1, 1 HtL x 1, 2 HtL x 1, n HtL x 2, 1 HtL x 2, 2 HtL x 2, n HtL x n,1 HtL x n,2 HtL x n, n HtL We can then express our solution as xhtl = XHtL c, where c = c 1 c 2 c n. In general a fundamental matrix for the system x' = Ax satisfies the corresponding matrix differential equation X' = AX. ü Example Write the given system in the matrix form x' = Ax+ f. x y = 2 x - 3 y + t = x - 2 y + sin t

3 CH_09_notes.nb 3 ü Example Determine whether the given vector functions are linearly dependent or linearly independent on H-, L. x 1 = 3 t t, x 2 = -t -t Theorem 4 - Representation of Solutions (Nonhomogeneous Case) Let x p be a particular solution to the nonhomogeneous system (9) x' HtL = AHtL xhtl + fhtl on the interval I, and let 8x 1,..., x n < be afundamental solution set on I for the corresponding homogeneous system x' HtL = A HtL xhtl. Then every solution to (9) on I can be expressed in the form (10) xhtl = x p HtL + c 1 x 1 HtL + c 2 x 2 HtL + + c n x n HtL where c 1, c 2,..., c n are constants. Approach to Solving Normal Systems 1. To determine a general solution to the nμn homogeneous system x' = Ax: (a) Find a fundamental solution set 8x 1,..., x n < that consists of n linearly independent solutions to the homogeneous system. (b) Form the linear combination x = X c = c 1 x 1 + c 2 x c n x n, where c = colhc 1, c 2,..., c n L is any constant vector and X 1...x n D is the fundamental matrix, to obtain a general solution. 2. To determine a general solution x p to the nonhomogeneous system x' = Ax+ f: (a) Find a particular solution x p to the nonhomogeneous system. (b) Form the sum of the particular solution and the general solution X c = c 1 x 1 + c 2 x c n x n to the corresponding homogeneous system in part 1, xhtl = x p + X c = x p + c 1 x 1 + c 2 x c n x n to obtain a general solution to the given system.

4 4 CH_09_notes.nb 9.5 Homogeneous Linear Systems with Constant Coefficients Homework: p # ü Introduction Similar to Chapter 4, we will first consider the homogeneous linear system with constant coefficients of the form (1) x' HtL = AxHtL where A is a (real) constant nμn matrix. As before, let us assume a solution has the form xhtl = r t u where r is a constant and u is a constant vector. Definition 3 - Eigenvalues and Eigenvectors Let A = Aa ij E be an nμn constant matrix. The eigenvalues of A are those (real or complex) numbers r for which HA - r IL u = 0 has at least one nontrivial solution u. The corresponding nontrivial solutions u are called the eigenvectors of A associated with r. ü Example Find the eigenvalues and eigenvectors of the given matrix A = Theorem 5 - n Linearly Independent Eigenvectors Suppose the nμn matrix A has n linearly independent eigenvectors u 1, u 2,..., u n. Let r i be the eigenvalue corresponding to u i. Then (12) 8 r 1 t u 1, r 2 t u 2,..., r n t u n < is a fundamental solution set (and XHtL r 1 t u 1 r 2 t u 2... r n t u n D is a fundamental matrix) on H-, L for the homogeneous system x' = Ax. Consequently, a general solution of x' = Ax is (13) xhtl = c 1 r 1 t u 1 + c 2 r 2 t u c n r n t u n, where c 1, c 2,..., c n are constants.

5 CH_09_notes.nb 5 ü Example Find a general solution of x' = Ax where A = Theorem 6 - Linear Independence of Eigenvectors If r 1,..., r m are distinct eigenvalues for the matrix A and u i is an eigenvector associated with r i, then u 1,..., u m are linearly independent. Corollary 1 - n Distinct Eigenvectors If the nμn constant matrix A has n distinct eigenvalues r 1,..., r n and u i is an eigenvector associated with r i, then 8 r 1 t u 1,..., r n t u n < is a fundamental solution set for the homogenous system x' = Ax. ü Example Solve the initial value problem x' = x, xh0l = Definition 4 - Real Symmetric Matrices A real symmetric matrix A is a matrix with real entries that satisfies A T = A. If A is an nμn real symmetric matrix, it is known that there always exist n linearly independent eigenvectors. If a matrix A is not symmetric, it is possible for A to have a repeated eigenvalue but not to have two linearly independent corresponding eigenvectors.

6 6 CH_09_notes.nb ü Example Find a general solution of x' = Ax where A =

7 CH_09_notes.nb Complex Eigenvalues Homework: p # ü Introduction Now we consider the case where the eigenvalues are complex conjugates, a Âb. Suppose r 1 =a+âb (where a and b are real numbers) is an eigenvalue of A, with corresponding eigenvector z = a +Âb, where a and b are real constant vectors. Complex Eigenvalues If the real matrix A has complex conjugate eigenvalues a Âb with corresponding eigenvectors a + Âb, then two linearly independent real vector solutions to x' HtL = AxHtL are (6) a t cos b t a - a t sin b t b (7) a t sin b t a + a t cos b t b ü Example Find a general solution of x' = Ax where A =

8 8 CH_09_notes.nb 9.7 Nonhomogeneous Linear Systems Homework: p # ü Undetermined Coefficients Now we consider the nonhomogeneous case, x' HtL = AxHtL + fhtl where A is an nμn constant matrix and the entries of fhtl are polynomials, exponential functions, sine and cosines, or finite sums and products of these functions. We will use the same procedure from Chapter 4 to find x p HtL. ü Example Find a general solution of x' = Ax+ f where A = and f = 6 t - 9 t - 5.

9 CH_09_notes.nb 9 ü Variation of Parameters Let XHtL be a fundamental matrix for the homogeneous system (6) x' HtL = AxHtL where the entries of A may be any continuous functions of t. The particular solution of the nonhomogeneous system (7) x' HtL = AxHtL + fhtl is given by (10) x p HtL = XHtL Ÿ X -1 HtL fhtl t and the general solution is given by (11) xhtl = XHtL c + XHtL Ÿ X -1 HtL fhtl t The solution to the initial value problem (12) x' HtL = AxHtL + fhtl, xht 0 L = x 0 is given by (13) xhtl = XHtL X -1 HtL x 0 + XHtL Ÿ t0 t X -1 HsL fhsl s ü Example Find a general solution of x' = Ax+ f where A = and f = t.