Problem 9. Show how to solve the following differential equation. You will need to use separation of variables Explain all your steps. Utz +41 = Uxx 02020, t50 tro 1(0₁6) = u(c₁ts=0 4 (2₁0) = f(₂), 14₂ (2₂0) =g6x) 02x5c- Ans. (an andnt + basim dot ) Sin Dix n-o an = ²/² √o b(x) Dinnux dx (x₁) = 12 bn = = = √₁ glus pinnix cla dn = (1²7) ² +4
Problem 9. Show how to solve the following differential equation. You will need to use separation of variables Explain all your steps. Utz +41 = Uxx 02020, t50 tro 1(0₁6) = u(c₁ts=0 4 (2₁0) = f(₂), 14₂ (2₂0) =g6x) 02x5c- Ans. (an andnt + basim dot ) Sin Dix n-o an = ²/² √o b(x) Dinnux dx (x₁) = 12 bn = = = √₁ glus pinnix cla dn = (1²7) ² +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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![**Problem 9**
**Objective:**
Show how to solve the following differential equation. You will need to use separation of variables. Explain all your steps.
**Equation:**
\[ U_{tt} + 4U = U_{xx} \]
**Conditions:**
For \(0 \leq x \leq c\), \(t \geq 0\):
- \(U(0,t) = U(c,t) = 0\), \(t \geq 0\)
- \(U(x,0) = f(x)\), \(U_t(x,0) = g(x)\), \(0 \leq x \leq c\)
**Solution:**
\[ U(x,t) = \sum_{n=0}^{\infty} \left( a_n \cos \lambda_n t + b_n \sin \lambda_n t \right) \sin \left(\frac{n \pi}{c} x \right) \]
**Coefficients:**
\[ a_n = \frac{2}{c} \int_0^c f(x) \sin \left(\frac{n \pi}{c} x \right) dx \]
\[ b_n = \frac{1}{\lambda_n} \frac{2}{c} \int_0^c g(x) \sin \left(\frac{n \pi}{c} x \right) dx \]
**Eigenvalue:**
\[ \lambda_n = \left(\frac{n \pi}{c}\right)^2 + 4 \]
**Note:**
This solution method involves expressing the solution \( U(x,t) \) as a Fourier series and determining the coefficients \( a_n \) and \( b_n \) using integrals over the given boundary conditions. The eigenvalue \( \lambda_n \) is modified by the constant term 4, which comes from the equation \( U_{tt} + 4U = U_{xx} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2Fa543dd23-f15c-4aad-946d-3ce6b77d6837%2Fah6e3a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 9**
**Objective:**
Show how to solve the following differential equation. You will need to use separation of variables. Explain all your steps.
**Equation:**
\[ U_{tt} + 4U = U_{xx} \]
**Conditions:**
For \(0 \leq x \leq c\), \(t \geq 0\):
- \(U(0,t) = U(c,t) = 0\), \(t \geq 0\)
- \(U(x,0) = f(x)\), \(U_t(x,0) = g(x)\), \(0 \leq x \leq c\)
**Solution:**
\[ U(x,t) = \sum_{n=0}^{\infty} \left( a_n \cos \lambda_n t + b_n \sin \lambda_n t \right) \sin \left(\frac{n \pi}{c} x \right) \]
**Coefficients:**
\[ a_n = \frac{2}{c} \int_0^c f(x) \sin \left(\frac{n \pi}{c} x \right) dx \]
\[ b_n = \frac{1}{\lambda_n} \frac{2}{c} \int_0^c g(x) \sin \left(\frac{n \pi}{c} x \right) dx \]
**Eigenvalue:**
\[ \lambda_n = \left(\frac{n \pi}{c}\right)^2 + 4 \]
**Note:**
This solution method involves expressing the solution \( U(x,t) \) as a Fourier series and determining the coefficients \( a_n \) and \( b_n \) using integrals over the given boundary conditions. The eigenvalue \( \lambda_n \) is modified by the constant term 4, which comes from the equation \( U_{tt} + 4U = U_{xx} \).
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