Use variation of parameters to solve the given nonhomogeneous system.
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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![**Solve the Nonhomogeneous System Using Variation of Parameters**
To solve the given nonhomogeneous system of differential equations, apply the method of variation of parameters.
The system is given by:
\[
\mathbf{X}' =
\begin{pmatrix}
0 & 4 \\
-1 & 5 \\
\end{pmatrix}
\mathbf{X} +
\begin{pmatrix}
4 \\
e^{-5t} \\
\end{pmatrix}
\]
Your task is to find \(\mathbf{X}(t)\).
**Explanation of the System:**
- The first term, \(\mathbf{X}'\), represents the derivative of the vector function \(\mathbf{X}(t)\).
- The matrix
\[
\begin{pmatrix}
0 & 4 \\
-1 & 5 \\
\end{pmatrix}
\]
is the coefficient matrix, which defines the relationship between the components of \(\mathbf{X}(t)\).
- The vector
\[
\begin{pmatrix}
4 \\
e^{-5t} \\
\end{pmatrix}
\]
is the nonhomogeneous part that influences the system externally.
**Objective:**
To solve for \(\mathbf{X}(t)\) by employing variation of parameters, determine the particular solution that satisfies this nonhomogeneous equation along with the complementary solution derived from the homogeneous counterpart. Fill in the solution in the provided box for \(\mathbf{X}(t)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74a50780-bdf2-45e2-b018-f4cc84bd693f%2F1172b0ea-b7da-474e-9c1d-145973e8544e%2Fwhk1eef_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solve the Nonhomogeneous System Using Variation of Parameters**
To solve the given nonhomogeneous system of differential equations, apply the method of variation of parameters.
The system is given by:
\[
\mathbf{X}' =
\begin{pmatrix}
0 & 4 \\
-1 & 5 \\
\end{pmatrix}
\mathbf{X} +
\begin{pmatrix}
4 \\
e^{-5t} \\
\end{pmatrix}
\]
Your task is to find \(\mathbf{X}(t)\).
**Explanation of the System:**
- The first term, \(\mathbf{X}'\), represents the derivative of the vector function \(\mathbf{X}(t)\).
- The matrix
\[
\begin{pmatrix}
0 & 4 \\
-1 & 5 \\
\end{pmatrix}
\]
is the coefficient matrix, which defines the relationship between the components of \(\mathbf{X}(t)\).
- The vector
\[
\begin{pmatrix}
4 \\
e^{-5t} \\
\end{pmatrix}
\]
is the nonhomogeneous part that influences the system externally.
**Objective:**
To solve for \(\mathbf{X}(t)\) by employing variation of parameters, determine the particular solution that satisfies this nonhomogeneous equation along with the complementary solution derived from the homogeneous counterpart. Fill in the solution in the provided box for \(\mathbf{X}(t)\).
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