Consider the following initial value problem. y" + 6y' + 25y = 8(t) + 6(t5a), y(0) = 1, y'(0) = 0 Find the Laplace transform of the differential equation. (Write your answer as a function of s.) £{y} = Use the Laplace transform to solve the given initial-value problem. ])+([ y(t) = t-r ・ 24(x - [ ]).(-
Consider the following initial value problem. y" + 6y' + 25y = 8(t) + 6(t5a), y(0) = 1, y'(0) = 0 Find the Laplace transform of the differential equation. (Write your answer as a function of s.) £{y} = Use the Laplace transform to solve the given initial-value problem. ])+([ y(t) = t-r ・ 24(x - [ ]).(-
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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Question
![**Initial Value Problem:**
Consider the following initial value problem.
\[ y'' + 6y' + 25y = \delta(t - \pi) + \delta(t - 5\pi), \quad y(0) = 1, \quad y'(0) = 0 \]
**Task:**
Find the Laplace transform of the differential equation. (Write your answer as a function of \(s\).)
\[ \mathcal{L}\{y\} = \boxed{\phantom{\quad\quad\quad}} \]
Use the Laplace transform to solve the given initial-value problem.
\[ y(t) = \left( \boxed{\phantom{\quad}} \right) + \left( \boxed{\phantom{\quad}} \right) \cdot u(t - \pi) + \left( \boxed{\phantom{\quad}} \right) \cdot u(t - \boxed{\phantom{\quad}}) \]
**Explanation:**
The problem involves the use of the Laplace transform to solve a differential equation with delta functions and initial conditions. You are required to express the solution \(y(t)\) involving unit step functions \(u(t-a)\) after finding the Laplace transform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74a50780-bdf2-45e2-b018-f4cc84bd693f%2Fb88ffcf3-9888-41ad-9d2c-44b0fe40ce01%2Fk0ef3y4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Initial Value Problem:**
Consider the following initial value problem.
\[ y'' + 6y' + 25y = \delta(t - \pi) + \delta(t - 5\pi), \quad y(0) = 1, \quad y'(0) = 0 \]
**Task:**
Find the Laplace transform of the differential equation. (Write your answer as a function of \(s\).)
\[ \mathcal{L}\{y\} = \boxed{\phantom{\quad\quad\quad}} \]
Use the Laplace transform to solve the given initial-value problem.
\[ y(t) = \left( \boxed{\phantom{\quad}} \right) + \left( \boxed{\phantom{\quad}} \right) \cdot u(t - \pi) + \left( \boxed{\phantom{\quad}} \right) \cdot u(t - \boxed{\phantom{\quad}}) \]
**Explanation:**
The problem involves the use of the Laplace transform to solve a differential equation with delta functions and initial conditions. You are required to express the solution \(y(t)\) involving unit step functions \(u(t-a)\) after finding the Laplace transform.
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