Please don’t copy the answer from any online sources. Please show ever steps clearly. Just answer 4
Please don’t copy the answer from any online sources. Please show ever steps clearly. Just answer 4
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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Please don’t copy the answer from any online sources. Please show ever steps clearly. Just answer 4
![### Problem 4
**Objective:**
Solve the initial value problem given below.
**Equation:**
\[
U_t = k U_{xx} + a x \quad (0 < x < c, \, t > 0)
\]
**With Conditions:**
- \( U(0, t) = 0 \)
- \( U(c, t) = 0 \)
- \( U(x, 0) = 0 \)
Here, \( k \) and \( a \) are positive constants.
**Useful Integral:**
\[
\int_0^c x(c^2 - x^2) \sin \left( \frac{n \pi}{c} x \right) \, dx = (-1)^{n+1} \frac{6c^4}{n^3 \pi^3}
\]
**Expression for \( x \):**
\[
x = \frac{2c}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin \left( \frac{n\pi}{c} x \right)
\]
**Answer:**
\[
U(x, t) = \frac{a}{6k} x(c^2 - x^2) + \frac{2c^3 a}{k \pi^3} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3} \sin \left( \frac{n \pi}{c} x \right) e^{-\frac{kn^2 \pi^2}{c^2} t}
\]
**Note:**
\[
\lim_{t \to \infty} U(x, t) = \frac{a}{6k} x(c^2 - x^2)
\]
---
### Problem 5
**Objective:**
Solve the following initial and boundary value problem.
**Equation:**
\[
U_t - 2kt U_{xx} = 0, \quad 0 < x < \pi, \, t > 0
\]
**Conditions:**
- \( U(0, t) = U(\pi, t) = 0, \quad t \geq 0 \)
- \( U(x, 0) = 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2Fbc537d07-9196-4a70-a99b-95e2612df3eb%2Fneuaruc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4
**Objective:**
Solve the initial value problem given below.
**Equation:**
\[
U_t = k U_{xx} + a x \quad (0 < x < c, \, t > 0)
\]
**With Conditions:**
- \( U(0, t) = 0 \)
- \( U(c, t) = 0 \)
- \( U(x, 0) = 0 \)
Here, \( k \) and \( a \) are positive constants.
**Useful Integral:**
\[
\int_0^c x(c^2 - x^2) \sin \left( \frac{n \pi}{c} x \right) \, dx = (-1)^{n+1} \frac{6c^4}{n^3 \pi^3}
\]
**Expression for \( x \):**
\[
x = \frac{2c}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin \left( \frac{n\pi}{c} x \right)
\]
**Answer:**
\[
U(x, t) = \frac{a}{6k} x(c^2 - x^2) + \frac{2c^3 a}{k \pi^3} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3} \sin \left( \frac{n \pi}{c} x \right) e^{-\frac{kn^2 \pi^2}{c^2} t}
\]
**Note:**
\[
\lim_{t \to \infty} U(x, t) = \frac{a}{6k} x(c^2 - x^2)
\]
---
### Problem 5
**Objective:**
Solve the following initial and boundary value problem.
**Equation:**
\[
U_t - 2kt U_{xx} = 0, \quad 0 < x < \pi, \, t > 0
\]
**Conditions:**
- \( U(0, t) = U(\pi, t) = 0, \quad t \geq 0 \)
- \( U(x, 0) = 2
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