Problem 3 solve litt U (0₁ t) = 0, U(IT₁ t) = 0 1₂(x,₁0) = 3/10(2x)-- 5pin (3x) 11(x₁0) = 0, Cl
Problem 3 solve litt U (0₁ t) = 0, U(IT₁ t) = 0 1₂(x,₁0) = 3/10(2x)-- 5pin (3x) 11(x₁0) = 0, Cl
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 3**
**Solve:**
\[ U_{tt} = a^2 U_{xx} \]
Boundary conditions:
\[ U(0, t) = 0, \quad U(\pi, t) = 0 \]
Initial conditions:
\[ U(x, 0) = 0, \quad U_t(x, 0) = 3 \sin(2x) - 5 \sin(3x) \]
(Note: \(\lambda < 0\) and \(\lambda = 0\) lead to the trivial solution, you can skip this case.)
**Answer:**
\[ U(x, t) = \frac{3}{2a} \sin(2x) \sin(2at) - \frac{5}{3a} \sin(3x) \sin(3at) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2Fc477e0bc-0e6c-4ab3-afcc-18ac73d3fde6%2Fh4p3wp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3**
**Solve:**
\[ U_{tt} = a^2 U_{xx} \]
Boundary conditions:
\[ U(0, t) = 0, \quad U(\pi, t) = 0 \]
Initial conditions:
\[ U(x, 0) = 0, \quad U_t(x, 0) = 3 \sin(2x) - 5 \sin(3x) \]
(Note: \(\lambda < 0\) and \(\lambda = 0\) lead to the trivial solution, you can skip this case.)
**Answer:**
\[ U(x, t) = \frac{3}{2a} \sin(2x) \sin(2at) - \frac{5}{3a} \sin(3x) \sin(3at) \]
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