[9] Find a 27-periodic solution u(x) of 5u"(x) + 3u(x) = f(x) where f(x) is a 27-periodic function such that 0 f(x) = sin(2x) 0 x < x <∞, -π≤x≤0 0≤x≤π/2 π/2 ≤ x ≤T.
[9] Find a 27-periodic solution u(x) of 5u"(x) + 3u(x) = f(x) where f(x) is a 27-periodic function such that 0 f(x) = sin(2x) 0 x < x <∞, -π≤x≤0 0≤x≤π/2 π/2 ≤ x ≤T.
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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![**Problem 9**
Find a \(2\pi\)-periodic solution \(u(x)\) of
\[ 5u''(x) + 3u(x) = f(x) \quad \text{for } -\infty < x < \infty, \]
where \( f(x) \) is a \(2\pi\)-periodic function defined by:
\[
f(x) =
\begin{cases}
0, & -\pi \leq x < 0 \\
\sin(2x), & 0 \leq x < \pi/2 \\
0, & \pi/2 \leq x \leq \pi.
\end{cases}
\]
This equation describes finding a periodic solution \( u(x) \) for a second-order inhomogeneous differential equation with given periodic boundary conditions on \( f(x) \). The function \( f(x) \) is piecewise defined within one period \( -\pi \leq x \leq \pi \).
### Explanation of Pieces:
1. **For \( -\pi \leq x < 0 \):** The function \( f(x) \) is zero.
2. **For \( 0 \leq x < \pi/2 \):** The function \( f(x) \) is given by \( \sin(2x) \).
3. **For \( \pi/2 \leq x \leq \pi \):** The function \( f(x) \) returns to zero.
### Summary:
The objective is to solve the differential equation for \( u(x) \) such that it is a \(2\pi\)-periodic function, considering the piecewise periodic nature of \( f(x) \).
This type of problem is often solved using Fourier series or other methods that leverage the periodic properties of the functions involved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2Fc2700033-9a05-4d9e-a86a-75a36ad49883%2F0nw223v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 9**
Find a \(2\pi\)-periodic solution \(u(x)\) of
\[ 5u''(x) + 3u(x) = f(x) \quad \text{for } -\infty < x < \infty, \]
where \( f(x) \) is a \(2\pi\)-periodic function defined by:
\[
f(x) =
\begin{cases}
0, & -\pi \leq x < 0 \\
\sin(2x), & 0 \leq x < \pi/2 \\
0, & \pi/2 \leq x \leq \pi.
\end{cases}
\]
This equation describes finding a periodic solution \( u(x) \) for a second-order inhomogeneous differential equation with given periodic boundary conditions on \( f(x) \). The function \( f(x) \) is piecewise defined within one period \( -\pi \leq x \leq \pi \).
### Explanation of Pieces:
1. **For \( -\pi \leq x < 0 \):** The function \( f(x) \) is zero.
2. **For \( 0 \leq x < \pi/2 \):** The function \( f(x) \) is given by \( \sin(2x) \).
3. **For \( \pi/2 \leq x \leq \pi \):** The function \( f(x) \) returns to zero.
### Summary:
The objective is to solve the differential equation for \( u(x) \) such that it is a \(2\pi\)-periodic function, considering the piecewise periodic nature of \( f(x) \).
This type of problem is often solved using Fourier series or other methods that leverage the periodic properties of the functions involved.
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