Consider the initial value problem 4 *(0) %3D 2 Form the complementary solution to the homogeneous equation. X.(t) = a1 + a2 help (formulas) help (matrices) Construct a particular solution by assuming the form x,(t) = e'a and solving for the undetermined constant vector a. help (formulas) i,(1) = help (matrices) Form the general solution x(t) = x.(t) + x,(t) and impose the initial condition to obtain the solution of the initial value problem. x1(t) x2(t) help (formulas) help (matrices)
Consider the initial value problem 4 *(0) %3D 2 Form the complementary solution to the homogeneous equation. X.(t) = a1 + a2 help (formulas) help (matrices) Construct a particular solution by assuming the form x,(t) = e'a and solving for the undetermined constant vector a. help (formulas) i,(1) = help (matrices) Form the general solution x(t) = x.(t) + x,(t) and impose the initial condition to obtain the solution of the initial value problem. x1(t) x2(t) help (formulas) help (matrices)
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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Question
100%
![**Consider the initial value problem**
\[
\vec{x}' = \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix} \vec{x} + \begin{bmatrix} e^t \\ 0 \end{bmatrix}, \quad \vec{x}(0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}.
\]
**Form the complementary solution to the homogeneous equation.**
\[
\vec{x}_c(t) = \alpha_1 \begin{bmatrix} \, \, \, \, \, \end{bmatrix} + \alpha_2 \begin{bmatrix} \, \, \, \, \, \end{bmatrix}
\]
[help (formulas)] [help (matrices)]
**Construct a particular solution by assuming the form** \(\vec{x}_p(t) = e^t \vec{a}\) and solving for the undetermined constant vector \(\vec{a}\).
\[
\vec{x}_p(t) = \begin{bmatrix} \, \, \, \, \, \end{bmatrix}
\]
[help (formulas)] [help (matrices)]
**Form the general solution** \(\vec{x}(t) = \vec{x}_c(t) + \vec{x}_p(t)\) **and impose the initial condition to obtain the solution of the initial value problem.**
\[
\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = \begin{bmatrix} \, \, \, \, \, \\ \, \, \, \, \, \end{bmatrix}
\]
[help (formulas)] [help (matrices)]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F411fdfa3-98d6-4c96-a283-c3e756a59776%2F1d00a948-6874-49eb-9daa-85a629520b1f%2Ffo030ur_processed.png&w=3840&q=75)
Transcribed Image Text:**Consider the initial value problem**
\[
\vec{x}' = \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix} \vec{x} + \begin{bmatrix} e^t \\ 0 \end{bmatrix}, \quad \vec{x}(0) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}.
\]
**Form the complementary solution to the homogeneous equation.**
\[
\vec{x}_c(t) = \alpha_1 \begin{bmatrix} \, \, \, \, \, \end{bmatrix} + \alpha_2 \begin{bmatrix} \, \, \, \, \, \end{bmatrix}
\]
[help (formulas)] [help (matrices)]
**Construct a particular solution by assuming the form** \(\vec{x}_p(t) = e^t \vec{a}\) and solving for the undetermined constant vector \(\vec{a}\).
\[
\vec{x}_p(t) = \begin{bmatrix} \, \, \, \, \, \end{bmatrix}
\]
[help (formulas)] [help (matrices)]
**Form the general solution** \(\vec{x}(t) = \vec{x}_c(t) + \vec{x}_p(t)\) **and impose the initial condition to obtain the solution of the initial value problem.**
\[
\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = \begin{bmatrix} \, \, \, \, \, \\ \, \, \, \, \, \end{bmatrix}
\]
[help (formulas)] [help (matrices)]
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