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The image shows a mathematical problem involving time functions and differential equations, written in Indonesian. Let me provide a translation and breakdown of the problem.

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**Problem Translation:**

**Task P02:**

Given a time function \( f(t) \) when \( t \geq 0 \):

a. \( f(t) = 5u(t) + 3e^{-2t} \)

b. \( \frac{d^2 x(t)}{dt^2} + 2\frac{dx(t)}{dt} = 2f(t) \)

Find the Laplace transform \( F(s) \) of the function \( f(t) \).

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### Solution Outline:

#### Part a: 
\( f(t) = 5u(t) + 3e^{-2t} \)

- \( u(t) \) is the Heaviside step function, which in Laplace transform is \( \frac{1}{s} \).
- The Laplace transform of \( e^{-2t} \) is \( \frac{1}{s+2} \).
- So, the Laplace transform of \( f(t) \) is:

\[
F(s) = 5 \cdot \frac{1}{s} + 3 \cdot \frac{1}{s+2}
\]

#### Part b: 
For the differential equation:

\[
\frac{d^2 x(t)}{dt^2} + 2\frac{dx(t)}{dt} = 2f(t)
\]

- Using the Laplace transform for derivatives:
  - \( \mathcal{L}\left( \frac{d^2x(t)}{dt^2} \right) = s^2 X(s) - s x(0) - \dot{x}(0) \)
  - \( \mathcal{L}\left( \frac{dx(t)}{dt} \right) = s X(s) - x(0) \)
  
  Let initial conditions \( x(0) = 0 \) and \( \dot{x}(0) = 0 \) for simplicity.
  
- The Laplace transform of the equation becomes:

\[
s^2 X(s) + 2s X(s) = 2F(s)
\]

You can substitute \( F(s) \) from part a.

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### Related Questions:
1. What is the Laplace transform of the second derivative term if initial conditions are non-zero?
2. How do you solve the transformed equation for \( X(s) \)?
3. What are the properties of the Heaviside function in Laplace transforms?
4. How does the step function \( u(t) \) influence the Laplace domain behavior of functions?
5. Can we apply inverse Laplace transform to get back to the time domain for \( x(t) \)?

### Tip:
When solving differential equations using Laplace transforms, always check the initial conditions carefully to handle the transformed terms correctly.

Jika terdapat sebuah fungsi waktu sebagai berikut: a. f(t) = 5u(t) + 3e^(-2t), b. d^2x(t)/dt^2 + 2dx(t)/dt = 2f(t), tentukan fungsi Laplace dari f(t).

Solve for the Laplace transform of a time function that includes a Heaviside step function and exponential decay term. Also, apply Laplace transforms to a second-order differential equation with given initial conditions.

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