Find a particular solution ₂ to by trying a solution of the form [a] -1 cos(4t). -15 0 5 cos(4t
Find a particular solution ₂ to by trying a solution of the form [a] -1 cos(4t). -15 0 5 cos(4t
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
![**Finding a Particular Solution to a Differential Equation**
**Objective:**
Find a particular solution \(\vec{x}_p\) to the given differential equation.
**Differential Equation:**
\[
\vec{x}'' =
\begin{bmatrix}
-14 & -1 \\
-1 & -15
\end{bmatrix}
\vec{x} +
\begin{bmatrix}
0 \\
5
\end{bmatrix}
\cos(4t)
\]
**Method:**
Assume a solution of the form:
\[
\begin{bmatrix}
c_1 \\
c_2
\end{bmatrix}
\cos(4t)
\]
**Steps:**
1. Substitute the assumed form into the differential equation.
2. Solve for the constants \(c_1\) and \(c_2\) to find the particular solution \(\vec{x}_p\).
These steps involve finding coefficients that satisfy both sides of the equation, ensuring that the assumed form indeed solves the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4100c518-1a8f-4898-88a2-d0f1a7669694%2F1025a44b-32f6-4dbb-b4f1-c45921f2c17f%2F3pqbroe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Particular Solution to a Differential Equation**
**Objective:**
Find a particular solution \(\vec{x}_p\) to the given differential equation.
**Differential Equation:**
\[
\vec{x}'' =
\begin{bmatrix}
-14 & -1 \\
-1 & -15
\end{bmatrix}
\vec{x} +
\begin{bmatrix}
0 \\
5
\end{bmatrix}
\cos(4t)
\]
**Method:**
Assume a solution of the form:
\[
\begin{bmatrix}
c_1 \\
c_2
\end{bmatrix}
\cos(4t)
\]
**Steps:**
1. Substitute the assumed form into the differential equation.
2. Solve for the constants \(c_1\) and \(c_2\) to find the particular solution \(\vec{x}_p\).
These steps involve finding coefficients that satisfy both sides of the equation, ensuring that the assumed form indeed solves the differential equation.
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