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The image contains problems from an exam in "Differential Equations" (Διαφορικές Εξισώσεις). Here's a breakdown of the questions:

1. **Θέμα 1ο (3 μονάδες):** Find the general solution of the differential equation:
   \[
   y''(x) + 2y'(x) + 2y(x) = 10 \cos(\alpha)
   \]

2. **Θέμα 2ο (2 μονάδες):** Solve the system of linear differential equations using the method of eigenvalues and eigenvectors:
   \[
   x'(t) = 3x(t) + 2y(t)
   \]
   \[
   y'(t) = y(t)
   \]

3. **Θέμα 3ο (2.5 μονάδες):** Solve the initial value problem using the Laplace transform:
   \[
   y''(t) - 3y'(t) + 2y(t) = 0
   \]
   with the initial conditions:
   \[
   y(0) = -2, \quad y'(0) = 2
   \]

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Find the general solution to the following differential equations and solve the given systems:

This webpage covers solving various types of differential equations, including second-order equations with trigonometric terms, using the Laplace Transform and Eigenvalue methods. Suitable for university-level students studying differential equations.

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