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1. Nonhomogeneous linear system. Consider the nonhomogeneous system x' = P(t)x+ g(t), g(t) = 0 where the 3 x 3 matrix P(t) is continuous for -∞ < t < ∞. Assume that the columns of matrix X(t) are the three solutions for the corresponding homogeneous system X(t) = [x₁(t), x₂(t),x3(t)] = ( 1 0 0 et -1 e¹(1+t) et ett et (a) Compute the Wronskian of X(t). Then verify the inverse of X(t) is 1 -e-t -1 e-t 1 (1 70). −e¯¹(1+t) e¯¹ (2+t), (b) Use the method of variation of parameters to find the solution of the original nonhomogeneous system subject to the initial condition x₁(0) = 0, x2(0) = 0, and x3(0) = 0. (No need to find P(t).)

1. Nonhomogeneous linear system. Consider the nonhomogeneous system x' = P(t)x+ g(t), g(t) = 0 where the 3 x 3 matrix P(t) is continuous for -∞ < t < ∞. Assume that the columns of matrix X(t) are the three solutions for the corresponding homogeneous system X(t) = [x₁(t), x₂(t),x3(t)] = ( 1 0 0 et -1 e¹(1+t) et ett et (a) Compute the Wronskian of X(t). Then verify the inverse of X(t) is 1 -e-t -1 e-t 1 (1 70). −e¯¹(1+t) e¯¹ (2+t), (b) Use the method of variation of parameters to find the solution of the original nonhomogeneous system subject to the initial condition x₁(0) = 0, x2(0) = 0, and x3(0) = 0. (No need to find P(t).)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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1. Nonhomogeneous linear system. Consider the nonhomogeneous system x' = P(t)x+ g(t), g(t) = 0 where the 3 x 3 matrix P(t) is continuous for -∞ < t < ∞. Assume that the columns of matrix X(t) are the three solutions for the corresponding homogeneous system X(t) = [x₁(t), x₂(t),x3(t)] = ( 1 0 et -1 e¹(1+t) -1 e-t 1 (1 (a) Compute the Wronskian of X(t). Then verify the inverse of X(t) is 1 -e-t 0 et ett et 70). −e¯¹(1+t) _e¯¹(2+t), (b) Use the method of variation of parameters to find the solution of the original nonhomogeneous system subject to the initial condition x₁(0) = 0, x2(0) = 0, and x3(0) = 0. (No need to find P(t).)
Transcribed Image Text:1. Nonhomogeneous linear system. Consider the nonhomogeneous system x' = P(t)x+ g(t), g(t) = 0 where the 3 x 3 matrix P(t) is continuous for -∞ < t < ∞. Assume that the columns of matrix X(t) are the three solutions for the corresponding homogeneous system X(t) = [x₁(t), x₂(t),x3(t)] = ( 1 0 et -1 e¹(1+t) -1 e-t 1 (1 (a) Compute the Wronskian of X(t). Then verify the inverse of X(t) is 1 -e-t 0 et ett et 70). −e¯¹(1+t) _e¯¹(2+t), (b) Use the method of variation of parameters to find the solution of the original nonhomogeneous system subject to the initial condition x₁(0) = 0, x2(0) = 0, and x3(0) = 0. (No need to find P(t).)
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