https://file.aiquickdraw.com/mathbot/file/1728949309777ll3fue5i.jpg

The image contains a system of differential equations with initial conditions, and the task is to solve it using the **Laplace transform**. Here's a breakdown of the problem:

1. **System of equations**:
   \[
   \begin{aligned}
   x'(t) - 2x(t) &= y(t), \\
   y'(t) - y(t) &= 1.
   \end{aligned}
   \]
   
2. **Initial conditions**:
   \[
   x(0) = 0, \quad y(0) = 0.
   \]

### Steps to solve using Laplace Transform:

1. **Apply the Laplace Transform** to both equations:
   - For \( x'(t) \), use \( \mathcal{L}\{x'(t)\} = sX(s) - x(0) \).
   - For \( y'(t) \), use \( \mathcal{L}\{y'(t)\} = sY(s) - y(0) \).

2. **Transform the system**:
   \[
   \begin{aligned}
   \mathcal{L}\{x'(t)\} - 2\mathcal{L}\{x(t)\} &= \mathcal{L}\{y(t)\}, \\
   \mathcal{L}\{y'(t)\} - \mathcal{L}\{y(t)\} &= \mathcal{L}\{1\}.
   \end{aligned}
   \]
   Substituting the initial conditions and known Laplace transforms:
   \[
   \begin{aligned}
   (sX(s)) - 2X(s) &= Y(s), \\
   (sY(s)) - Y(s) &= \frac{1}{s}.
   \end{aligned}
   \]

3. **Solve the second equation** for \( Y(s) \):
   \[
   sY(s) - Y(s) = \frac{1}{s} \implies Y(s)(s - 1) = \frac{1}{s} \implies Y(s) = \frac{1}{s(s - 1)}.
   \]

4. **Solve the first equation** for \( X(s) \):
   \[
   sX(s) - 2X(s) = Y(s) \implies X(s)(s - 2) = Y(s) = \frac{1}{s(s - 1)}.
   \]
   Therefore,
   \[
   X(s) = \frac{1}{s(s - 1)(s - 2)}.
   \]

5. **Inverse Laplace Transform**:
   - The function \( Y(s) = \frac{1}{s(s - 1)} \) can be decomposed using partial fractions:
     \[
     \frac{1}{s(s - 1)} = \frac{A}{s} + \frac{B}{s - 1},
     \]
     which gives \( A = 1 \) and \( B = -1 \), leading to:
     \[
     Y(s) = \frac{1}{s} - \frac{1}{s - 1}.
     \]
     The inverse Laplace transform is:
     \[
     y(t) = 1 - e^t.
     \]
   - For \( X(s) \), use partial fraction decomposition and the inverse Laplace transform.

Would you like me to continue with the detailed decomposition of \( X(s) \), or do you have any specific questions? Here are some related questions:

1. Would you like to see the detailed inverse Laplace steps for \( X(s) \)?
2. Do you need a step-by-step explanation of the partial fraction decomposition?
3. How does the Laplace transform help in solving differential equations?
4. Do you need a refresher on solving linear systems of differential equations?
5. Would you like to explore specific applications of the Laplace transform in engineering?

**Tip:** The Laplace transform is particularly useful for handling systems with initial conditions, making it easier to switch to the frequency domain and solve algebraically.

Resolver el sistema con transformada de Laplace con valores iniciales:
1. x'(t) - 2x(t) = y(t), y'(t) - y(t) = 1.
2. Condiciones iniciales: x(0) = 0, y(0) = 0.

Learn how to solve a system of differential equations using the Laplace transform. The problem includes equations with initial conditions, demonstrating steps for applying the Laplace transform and finding the inverse Laplace for the solution.

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