Consider the following initial value problem: Edit y" + 8y' + 15y = d(t – 5) + u10(t); y(0) = 0, y(0) 4 a) Find the solution y(t). y(t) 8. -3t - e 1 -3(t-5) e 1 -5(t-5) u(t) X 1 e5(t–10) 10 1 -3(t-10) e ua(t) 6. 15 where c= and d =10
Consider the following initial value problem: Edit y" + 8y' + 15y = d(t – 5) + u10(t); y(0) = 0, y(0) 4 a) Find the solution y(t). y(t) 8. -3t - e 1 -3(t-5) e 1 -5(t-5) u(t) X 1 e5(t–10) 10 1 -3(t-10) e ua(t) 6. 15 where c= and d =10
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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![**Initial Value Problem and Its Solution**
Consider the following initial value problem:
\[ y'' + 8y' + 15y = \delta(t - 5) + u_{10}(t); \quad y(0) = 0, \quad y'(0) = \frac{1}{4} \]
**a) Find the solution \( y(t) \).**
The solution to the problem is given by:
\[ y(t) = \frac{1}{8} \left( e^{-3t} - e^{-5t} \right) \]
\[ + \left( \frac{1}{2} \left( e^{-3(t-5)} - \frac{1}{2} e^{-5(t-5)} \right) \right) u_c(t) \]
\[ + \left( -\frac{1}{6} e^{-3(t-10)} + \frac{1}{10} e^{-5(t-10)} + \frac{1}{15} \right) u_d(t) \]
where \( c = 5 \) and \( d = 10 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F426fc7fa-6ee2-46f8-8cce-a78cc8035a57%2Fb8d1d8f8-8491-42af-ad89-1ad29c623afb%2Fknumzn4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Initial Value Problem and Its Solution**
Consider the following initial value problem:
\[ y'' + 8y' + 15y = \delta(t - 5) + u_{10}(t); \quad y(0) = 0, \quad y'(0) = \frac{1}{4} \]
**a) Find the solution \( y(t) \).**
The solution to the problem is given by:
\[ y(t) = \frac{1}{8} \left( e^{-3t} - e^{-5t} \right) \]
\[ + \left( \frac{1}{2} \left( e^{-3(t-5)} - \frac{1}{2} e^{-5(t-5)} \right) \right) u_c(t) \]
\[ + \left( -\frac{1}{6} e^{-3(t-10)} + \frac{1}{10} e^{-5(t-10)} + \frac{1}{15} \right) u_d(t) \]
where \( c = 5 \) and \( d = 10 \).
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