t X = X₁ + X₂ = eAtc + eAt CA: ['e to X(t) = Find the general solution of the given system. ở)x + (sinh(Đ)) X cosh(t) X'= e-ASF(s) ds 0 (5)
t X = X₁ + X₂ = eAtc + eAt CA: ['e to X(t) = Find the general solution of the given system. ở)x + (sinh(Đ)) X cosh(t) X'= e-ASF(s) ds 0 (5)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Equation for General Solution**
The general solution of a linear differential equation system is given by:
\[ \mathbf{X} = \mathbf{X}_c + \mathbf{X}_p = e^{At}\mathbf{C} + e^{At} \int_{t_0}^{t} e^{-As} \mathbf{F}(s) \, ds \]
**Problem Statement**
Find the general solution of the given system:
\[
\mathbf{X}' = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \mathbf{X} + \begin{pmatrix} \sinh(t) \\ \cosh(t) \end{pmatrix}
\]
**Solution Form**
\[ \mathbf{X}(t) = \boxed{} \]
**Explanation:**
- \(\mathbf{X} = \mathbf{X}_c + \mathbf{X}_p\) represents the complete solution, consisting of the complementary solution \(\mathbf{X}_c\) and the particular solution \(\mathbf{X}_p\).
- The matrix \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) is the system matrix for the homogeneous part.
- The vector \(\begin{pmatrix} \sinh(t) \\ \cosh(t) \end{pmatrix}\) functions as the non-homogeneous part of the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74a50780-bdf2-45e2-b018-f4cc84bd693f%2F1f28764a-4141-4de8-812d-87dd0c9570bd%2F2puvdzs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Equation for General Solution**
The general solution of a linear differential equation system is given by:
\[ \mathbf{X} = \mathbf{X}_c + \mathbf{X}_p = e^{At}\mathbf{C} + e^{At} \int_{t_0}^{t} e^{-As} \mathbf{F}(s) \, ds \]
**Problem Statement**
Find the general solution of the given system:
\[
\mathbf{X}' = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \mathbf{X} + \begin{pmatrix} \sinh(t) \\ \cosh(t) \end{pmatrix}
\]
**Solution Form**
\[ \mathbf{X}(t) = \boxed{} \]
**Explanation:**
- \(\mathbf{X} = \mathbf{X}_c + \mathbf{X}_p\) represents the complete solution, consisting of the complementary solution \(\mathbf{X}_c\) and the particular solution \(\mathbf{X}_p\).
- The matrix \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) is the system matrix for the homogeneous part.
- The vector \(\begin{pmatrix} \sinh(t) \\ \cosh(t) \end{pmatrix}\) functions as the non-homogeneous part of the system.
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