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).)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ea01508-f9b6-4a19-b038-56e12c298daf%2F9d7b3736-53b2-4577-8bf2-cbce2e5d037f%2Fsnklmynq_processed.png&w=3840&q=75)
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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