Solve the initial value problem by Laplace transform. The given differential equation is:\[ y'' + y = -2 \sin t + 10 \delta(t - \pi), \]with initial conditions:\[ y(0) = 0, \]\[ y'(0) = 1. \]### Explanation:- **Equation**: This is a second-order linear differential equation with a delta function as an input. The delta function, denoted $\delta(t - \pi)$, represents an impulse applied at $t = \pi$.- **Terms**: - $y''$ is the second derivative of $y$ with respect to $t$. - $y$ represents the function being studied. - $-2 \sin t$ is a sinusoidal forcing term. - The term $10 \delta(t - \pi)$ indicates an impulse of magnitude 10 applied at time $t = \pi$. - **Initial Conditions**: - $y(0) = 0$ specifies that the function $y$ is 0 at $t = 0$. - $y'(0) = 1$ specifies that the derivative of $y$ with respect to $t$ is 1 at $t = 0$.This type of equation might be encountered in the context of mechanical vibrations or electrical circuits, where an external force or input (represented by the sine and delta functions) affects the system's behavior over time.

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Answered: y" + y = -2 sin t + 10 8(t – 7), y(0) =…

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y" + y = -2 sin t + 10 8(t – 7), y(0) = 0, y'(0) = 1.

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

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Solve the initial value problem by Laplace transform.

The given differential equation is:

\[ y'' + y = -2 \sin t + 10 \delta(t - \pi), \]

with initial conditions:

\[ y(0) = 0, \]
\[ y'(0) = 1. \]

### Explanation:

- **Equation**: This is a second-order linear differential equation with a delta function as an input. The delta function, denoted $\delta(t - \pi)$, represents an impulse applied at $t = \pi$.
- **Terms**:
- $y''$ is the second derivative of $y$ with respect to $t$.
- $y$ represents the function being studied.
- $-2 \sin t$ is a sinusoidal forcing term.
- The term $10 \delta(t - \pi)$ indicates an impulse of magnitude 10 applied at time $t = \pi$.

- **Initial Conditions**:
- $y(0) = 0$ specifies that the function $y$ is 0 at $t = 0$.
- $y'(0) = 1$ specifies that the derivative of $y$ with respect to $t$ is 1 at $t = 0$.

This type of equation might be encountered in the context of mechanical vibrations or electrical circuits, where an external force or input (represented by the sine and delta functions) affects the system's behavior over time.

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