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The given differential equation is:

\[
x''(t) - 4x(t) = t
\]
with initial conditions:
\[
x(0) = -1, \quad x'(0) = 1
\]

We are asked to solve this differential equation using the Laplace transform. Let's break this down step-by-step:

### Step 1: Apply the Laplace Transform
First, apply the Laplace transform to both sides of the equation. Recall that the Laplace transform of a derivative is given by:

- \( \mathcal{L}\{x''(t)\} = s^2 X(s) - s x(0) - x'(0) \)
- \( \mathcal{L}\{x'(t)\} = s X(s) - x(0) \)
- \( \mathcal{L}\{x(t)\} = X(s) \)

Applying the Laplace transform to the entire equation:

\[
\mathcal{L}\{x''(t) - 4x(t)\} = \mathcal{L}\{t\}
\]

This results in:

\[
s^2 X(s) - s(-1) - 1 - 4 X(s) = \frac{1}{s^2}
\]

Simplifying:

\[
(s^2 - 4) X(s) + s + 1 = \frac{1}{s^2}
\]

### Step 2: Solve for \( X(s) \)
Now, isolate \( X(s) \):

\[
X(s) = \frac{1}{s^2 (s^2 - 4)} - \frac{s + 1}{s^2 - 4}
\]

### Step 3: Perform Inverse Laplace Transform
Next, decompose the expression for \( X(s) \) using partial fraction decomposition, and apply the inverse Laplace transform to find \( x(t) \).

Would you like a detailed continuation of the steps, including the partial fraction decomposition and inverse Laplace transform, or do you have specific questions about the process so far?

### Related Questions:
1. What is the general solution of a second-order differential equation using the Laplace transform?
2. How do the initial conditions affect the solution in Laplace domain?
3. How do you decompose a rational function into partial fractions?
4. What are the main properties of the Laplace transform that simplify differential equations?
5. Can this method be used for non-linear differential equations?

### Tip:
Always check the initial conditions carefully, as they play a crucial role in simplifying the Laplace-transformed equation.

Resuelva la siguiente Ecuación Diferencial mediante transformada de Laplace: x''(t) - 4x(t) = t, con condiciones iniciales: x(0) = -1 y x'(0) = 1

Learn how to solve the second-order differential equation x''(t) - 4x(t) = t with initial conditions x(0) = -1 and x'(0) = 1 using the Laplace transform. This problem involves key mathematical concepts such as differential equations, the Laplace transform, and partial fraction decomposition. Suitable for advanced calculus students.

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