Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 - 24e-1 -9e-21 060 -5 x'= 10 1x, x(0)= 8e3t x(t)= 1 1 0 -4e-t 3e-21 3t 4e -2 -20e-t 3e-21 3t 4e 060 (a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A = 10 1 validate the following identities and write the column vector that equals each side of the equation. 1 1 0 ×₁' = Ax₁ = x2' = Ax₂ = x3' = Ax3 (b) Next, compute the Wronskian of X(t). W[x()()()]= Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix. (c) Find x(t) = ☐ (d) x(t) = x(t)x¯1 (0)x₁ =
Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 - 24e-1 -9e-21 060 -5 x'= 10 1x, x(0)= 8e3t x(t)= 1 1 0 -4e-t 3e-21 3t 4e -2 -20e-t 3e-21 3t 4e 060 (a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A = 10 1 validate the following identities and write the column vector that equals each side of the equation. 1 1 0 ×₁' = Ax₁ = x2' = Ax₂ = x3' = Ax3 (b) Next, compute the Wronskian of X(t). W[x()()()]= Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix. (c) Find x(t) = ☐ (d) x(t) = x(t)x¯1 (0)x₁ =
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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![Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 -
24e-1 -9e-21
060
-5
x'=
10 1x,
x(0)=
8e3t
x(t)=
1 1 0
-4e-t
3e-21
3t
4e
-2
-20e-t
3e-21
3t
4e
060
(a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A =
10 1
validate the following identities and write the column vector that equals each side of the equation.
1 1 0
×₁' = Ax₁ =
x2' = Ax₂ =
x3' = Ax3
(b) Next, compute the Wronskian of X(t).
W[x()()()]=
Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix.
(c) Find x(t) = ☐
(d) x(t) = x(t)x¯1 (0)x₁ =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf84dde2-f77b-4203-b744-f3414691451c%2Fe28c96b8-6151-482c-9392-3b77ae90ce3e%2Fel3hdnm_processed.png&w=3840&q=75)
Transcribed Image Text:Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 -
24e-1 -9e-21
060
-5
x'=
10 1x,
x(0)=
8e3t
x(t)=
1 1 0
-4e-t
3e-21
3t
4e
-2
-20e-t
3e-21
3t
4e
060
(a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A =
10 1
validate the following identities and write the column vector that equals each side of the equation.
1 1 0
×₁' = Ax₁ =
x2' = Ax₂ =
x3' = Ax3
(b) Next, compute the Wronskian of X(t).
W[x()()()]=
Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix.
(c) Find x(t) = ☐
(d) x(t) = x(t)x¯1 (0)x₁ =
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