Solve y + 2y + 10y -90( 15), y(0) = 1, y (0)
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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Please walk me through how to do this problem involving the delta function in a second order
![The equation depicted is as follows:
\[
y(t) = e^{-t} \cos 3t + \frac{4}{3} e^{-t} \sin 3t - 3 u(t-10) e^{-(t-10)} \sin 3(t-10) + 4 u(t-15) e^{-(t-15)} \sin 3(t-15)
\]
Explanation:
- This is a function \( y(t) \) that involves exponential decay and trigonometric oscillations.
- The terms \( e^{-t} \cos 3t \) and \( \frac{4}{3} e^{-t} \sin 3t \) represent a combination of exponentially decaying cosine and sine waves.
- The Heaviside step function \( u(t-10) \) activates the third term at \( t = 10 \).
- The term \( -3 u(t-10) e^{-(t-10)} \sin 3(t-10) \) indicates an exponentially decaying sine wave starting at \( t = 10 \).
- Similarly, \( u(t-15) \) activates the fourth term at \( t = 15 \).
- The term \( 4 u(t-15) e^{-(t-15)} \sin 3(t-15) \) describes an exponentially decaying sine wave starting at \( t = 15 \).
This function likely models a physical system's response, such as a damped harmonic oscillator subjected to sudden forces at \( t = 10 \) and \( t = 15 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F752c80f7-6a34-4004-b4db-3d79d0db3666%2Fa9ea104b-04cb-4f1d-835d-7d9c74652187%2F0htjdi_processed.png&w=3840&q=75)
Transcribed Image Text:The equation depicted is as follows:
\[
y(t) = e^{-t} \cos 3t + \frac{4}{3} e^{-t} \sin 3t - 3 u(t-10) e^{-(t-10)} \sin 3(t-10) + 4 u(t-15) e^{-(t-15)} \sin 3(t-15)
\]
Explanation:
- This is a function \( y(t) \) that involves exponential decay and trigonometric oscillations.
- The terms \( e^{-t} \cos 3t \) and \( \frac{4}{3} e^{-t} \sin 3t \) represent a combination of exponentially decaying cosine and sine waves.
- The Heaviside step function \( u(t-10) \) activates the third term at \( t = 10 \).
- The term \( -3 u(t-10) e^{-(t-10)} \sin 3(t-10) \) indicates an exponentially decaying sine wave starting at \( t = 10 \).
- Similarly, \( u(t-15) \) activates the fourth term at \( t = 15 \).
- The term \( 4 u(t-15) e^{-(t-15)} \sin 3(t-15) \) describes an exponentially decaying sine wave starting at \( t = 15 \).
This function likely models a physical system's response, such as a damped harmonic oscillator subjected to sudden forces at \( t = 10 \) and \( t = 15 \).
![**Problem 43**
Solve the differential equation:
\[ y'' + 2y' + 10y = -9\delta(t - 10) + 12\delta(t - 15) \]
with initial conditions \( y(0) = 1 \) and \( y'(0) = 3 \).
---
**Description:**
This problem involves solving a second-order linear differential equation with constant coefficients. The right side of the equation contains Dirac delta functions, \(\delta(t - 10)\) and \(\delta(t - 15)\), which represent impulse inputs applied at times \(t = 10\) and \(t = 15\), respectively. The equation is subject to the initial conditions \( y(0) = 1 \) and \( y'(0) = 3 \).
The goal is to determine the function \(y(t)\) that satisfies the differential equation and respects the given conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F752c80f7-6a34-4004-b4db-3d79d0db3666%2Fa9ea104b-04cb-4f1d-835d-7d9c74652187%2F2loimba_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 43**
Solve the differential equation:
\[ y'' + 2y' + 10y = -9\delta(t - 10) + 12\delta(t - 15) \]
with initial conditions \( y(0) = 1 \) and \( y'(0) = 3 \).
---
**Description:**
This problem involves solving a second-order linear differential equation with constant coefficients. The right side of the equation contains Dirac delta functions, \(\delta(t - 10)\) and \(\delta(t - 15)\), which represent impulse inputs applied at times \(t = 10\) and \(t = 15\), respectively. The equation is subject to the initial conditions \( y(0) = 1 \) and \( y'(0) = 3 \).
The goal is to determine the function \(y(t)\) that satisfies the differential equation and respects the given conditions.
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