https://file.aiquickdraw.com/mathbot/file/1730437057587sifolzps.jpg

can you solve this question read the question carefully and give your answers

Let's go through the problem step-by-step.

### Step 1: Taking the Laplace Transform

Given the differential equation:
\[
y'' + 4y = f(t),
\]
where:
\[
f(t) = 
\begin{cases} 
\sin(\pi t), & 0 \leq t < 1 \\
0, & 1 \leq t 
\end{cases}
\]
with initial conditions \( y(0) = 0 \) and \( y'(0) = 0 \).

1. **Laplace Transform of the Left Side**:
   - For \( y'' \), the Laplace transform is \( s^2 Y(s) - sy(0) - y'(0) \).
   - For \( 4y \), the Laplace transform is \( 4Y(s) \).
   
   Using the initial conditions \( y(0) = 0 \) and \( y'(0) = 0 \), we have:
   \[
   \mathcal{L}(y'') + 4 \mathcal{L}(y) = s^2 Y(s) + 4Y(s).
   \]
   Factoring out \( Y(s) \), this becomes:
   \[
   (s^2 + 4) Y(s) = \mathcal{L}(f(t)).
   \]

2. **Laplace Transform of the Right Side**:
   Since \( f(t) \) is defined piecewise, we use the Laplace transform for each interval:
   \[
   f(t) = \sin(\pi t) \cdot u_c(t) \quad \text{where} \quad u_c(t) = 
   \begin{cases} 
   1, & 0 \leq t < 1 \\
   0, & 1 \leq t.
   \end{cases}
   \]
   Using the unit step function to represent this, the Laplace transform is:
   \[
   \mathcal{L}\left(\sin(\pi t) u_c(t)\right) = \frac{\pi}{(s^2 + \pi^2)(s^2 + 4)}.
   \]

So, we find:
\[
Y(s) = \frac{\pi}{(s^2 + \pi^2)(s^2 + 4)}.
\]

### Step 2: Finding the Inverse Laplace Transform

We have:
\[
Y(s) = \frac{\pi}{(s^2 + \pi^2)(s^2 + 4)}.
\]
Using partial fraction decomposition, we can rewrite \( Y(s) \) as:
\[
Y(s) = \frac{\pi}{\pi^2 - 4} \left( \frac{1}{s^2 + 4} - \frac{1}{s^2 + \pi^2} \right).
\]
Now we take the inverse Laplace transform of each term separately.

1. **Inverse Laplace of \( \frac{1}{s^2 + 4} \)**:
   \[
   \mathcal{L}^{-1} \left( \frac{1}{s^2 + 4} \right) = \sin(2t).
   \]

2. **Inverse Laplace of \( \frac{1}{s^2 + \pi^2} \)**:
   \[
   \mathcal{L}^{-1} \left( \frac{1}{s^2 + \pi^2} \right) = \sin(\pi t).
   \]

Combining these results, we have:
\[
y(t) = \frac{\pi}{\pi^2 - 4} \left( \sin(2t) - \sin(\pi t) \right).
\]

### Final Answer:

1. **Laplace Transform Solution \( Y(s) \)**:
   \[
   Y(s) = \frac{\pi}{(s^2 + \pi^2)(s^2 + 4)}.
   \]

2. **Inverse Transform Solution \( y(t) \)**:
   \[
   y(t) = \frac{\pi}{\pi^2 - 4} \left( \sin(2t) - \sin(\pi t) \right).
   \]

---

Would you like further details on any specific step, or do you have any questions?

---

### Related Questions:
1. How can we use the Laplace transform to solve other differential equations with different initial conditions?
2. How would the solution change if \( f(t) \) had a different piecewise definition?
3. What are some alternative methods for solving this differential equation without using Laplace transforms?
4. How does the unit step function help in solving piecewise-defined functions with the Laplace transform?
5. Can you explain the partial fraction decomposition in more detail?

### Tip:
When solving piecewise functions with the Laplace transform, using the unit step function simplifies the representation and allows for an easier calculation.

Take the Laplace transform of the following initial value and solve for Y(s) = L{y(t)}: y'' + 4y = {sin(πt), 0 ≤ t < 1, 0, 1 ≤ t}, y(0) = 0, y'(0) = 0. Next, take the inverse transform of Y(s) to get y(t). Use step(t-c) for uc(t).

Learn how to solve a differential equation using the Laplace transform with initial conditions y(0) = 0 and y'(0) = 0, and a piecewise function. The problem involves applying the Laplace and Inverse Laplace transforms, handling piecewise functions, and using partial fraction decomposition for the solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation