Express the solution of the given initial value problem in terms of a convolution integral y" + 12y' + 32y = cos(at); y(0) = 1, y'(0) = 0 y(t) = X+/ (--) - e-) cos(a 7) dr/ -2 e-8t +e-4t -4(t-T) -8(t-r)
Express the solution of the given initial value problem in terms of a convolution integral y" + 12y' + 32y = cos(at); y(0) = 1, y'(0) = 0 y(t) = X+/ (--) - e-) cos(a 7) dr/ -2 e-8t +e-4t -4(t-T) -8(t-r)
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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![**Title: Solving Initial Value Problems Using Convolution Integrals**
**Problem Statement:**
Express the solution of the given initial value problem in terms of a convolution integral:
\[ y'' + 12y' + 32y = \cos(at); \quad y(0) = 1, \quad y'(0) = 0 \]
**Solution:**
The function \( y(t) \) is given by:
\[
y(t) = \left[ -2e^{-8t} + e^{-4t} \right] + \int_{0}^{t} \left[ \frac{1}{4} \left( e^{-4(t-\tau)} - e^{-8(t-\tau)} \right) \cos(a\tau) \right] d\tau
\]
**Explanation:**
1. **Homogeneous Solution:**
- The term \( -2e^{-8t} + e^{-4t} \) represents the homogeneous solution of the differential equation. These are the solutions obtained from solving the characteristic equation associated with the differential equation without the forcing function.
2. **Particular Solution using Convolution Integral:**
- The integral \(\int_{0}^{t} [...] d\tau\) represents the particular solution of the differential equation.
- Inside the integral, \(\frac{1}{4} \left( e^{-4(t-\tau)} - e^{-8(t-\tau)} \right)\) is the impulse response of the system.
- \(\cos(a\tau)\) is the non-homogeneous term, i.e., the forcing function.
- The convolution of the impulse response with the forcing function gives the particular solution.
This mixed approach, using both the homogeneous solution and a convolution integral for the particular solution, provides the complete solution to the initial value problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf5833d1-6930-42b3-a1f0-37b0b1113f5c%2F670e0144-5902-48de-a6cd-40e8e38cde83%2F5iwldtf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Initial Value Problems Using Convolution Integrals**
**Problem Statement:**
Express the solution of the given initial value problem in terms of a convolution integral:
\[ y'' + 12y' + 32y = \cos(at); \quad y(0) = 1, \quad y'(0) = 0 \]
**Solution:**
The function \( y(t) \) is given by:
\[
y(t) = \left[ -2e^{-8t} + e^{-4t} \right] + \int_{0}^{t} \left[ \frac{1}{4} \left( e^{-4(t-\tau)} - e^{-8(t-\tau)} \right) \cos(a\tau) \right] d\tau
\]
**Explanation:**
1. **Homogeneous Solution:**
- The term \( -2e^{-8t} + e^{-4t} \) represents the homogeneous solution of the differential equation. These are the solutions obtained from solving the characteristic equation associated with the differential equation without the forcing function.
2. **Particular Solution using Convolution Integral:**
- The integral \(\int_{0}^{t} [...] d\tau\) represents the particular solution of the differential equation.
- Inside the integral, \(\frac{1}{4} \left( e^{-4(t-\tau)} - e^{-8(t-\tau)} \right)\) is the impulse response of the system.
- \(\cos(a\tau)\) is the non-homogeneous term, i.e., the forcing function.
- The convolution of the impulse response with the forcing function gives the particular solution.
This mixed approach, using both the homogeneous solution and a convolution integral for the particular solution, provides the complete solution to the initial value problem.
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