Use the Laplace transform to solve the given initial-value problem. y" + 6y' + 34y = 5(t – x) + 6(t - 5x), y(0) = 1, y'(0) = 0 ) •([ ]) «(e - «) + ([ ) «(e - y(t) = +
Use the Laplace transform to solve the given initial-value problem. y" + 6y' + 34y = 5(t – x) + 6(t - 5x), y(0) = 1, y'(0) = 0 ) •([ ]) «(e - «) + ([ ) «(e - y(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 Statement**: Use the Laplace transform to solve the given initial-value problem.
**Equation**:
\[ y'' + 6y' + 34y = \delta(t - \pi) + \delta(t - 5\pi) \]
**Initial Conditions**:
\[ y(0) = 1, \quad y'(0) = 0 \]
**Solution Form**:
\[ y(t) = \left( \,\, \boxed{\phantom{solution}} \,\, \right) + \left( \,\, \boxed{\phantom{solution}} \,\, \right) \cdot u(t - \pi) + \left( \,\, \boxed{\phantom{solution}} \,\, \right) \cdot u(t - \boxed{\phantom{solution}}) \]
*Note*: The equation contains the Dirac delta function \(\delta\) at \(t = \pi\) and \(t = 5\pi\), and incorporates Heaviside step functions \(u(t - \pi)\) and \(u(t - 5\pi)\) in the solution. The boxes represent placeholders for components of the solution derived using Laplace transforms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20fa932-311a-46cd-86ab-78d74c804e39%2Fafbdcdd4-f913-44b3-bd71-4613354fd9d1%2F86t9cj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**: Use the Laplace transform to solve the given initial-value problem.
**Equation**:
\[ y'' + 6y' + 34y = \delta(t - \pi) + \delta(t - 5\pi) \]
**Initial Conditions**:
\[ y(0) = 1, \quad y'(0) = 0 \]
**Solution Form**:
\[ y(t) = \left( \,\, \boxed{\phantom{solution}} \,\, \right) + \left( \,\, \boxed{\phantom{solution}} \,\, \right) \cdot u(t - \pi) + \left( \,\, \boxed{\phantom{solution}} \,\, \right) \cdot u(t - \boxed{\phantom{solution}}) \]
*Note*: The equation contains the Dirac delta function \(\delta\) at \(t = \pi\) and \(t = 5\pi\), and incorporates Heaviside step functions \(u(t - \pi)\) and \(u(t - 5\pi)\) in the solution. The boxes represent placeholders for components of the solution derived using Laplace transforms.
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