The document you've provided contains two questions related to solving differential equations.

Question 1 (21 Marks)

You are asked to solve three different second-order differential equations:

a) $2 \frac{d^2 y}{dx^2} - 7 \frac{dy}{dx} - 4y = e^{3x}$

This is a non-homogeneous linear differential equation. The solution will involve both a complementary solution (for the homogeneous equation) and a particular solution for the non-homogeneous part.

b) $\frac{d^2 y}{dx^2} + 4 \frac{dy}{dx} + 4y = 2 \cos^2 x$

This is also a non-homogeneous second-order differential equation. The non-homogeneous term involves $\cos^2 x$, which can be rewritten using a trigonometric identity.

c) $\frac{d^2 y}{dx^2} + \frac{dy}{dx} - 2y = 2 \cosh 2x$

This equation is non-homogeneous as well, with the non-homogeneous term $2 \cosh 2x$, which can be treated using standard methods for hyperbolic functions.

Question 2 (9 Marks)

This question involves solving the second-order differential equation:

$\frac{d^2 x}{dt^2} + 4 \frac{dx}{dt} + 3x = e^{-3t}$

Given the initial conditions:

• $x(0) = \frac{1}{2}$
• $\frac{dx}{dt}(0) = -2$

This is a non-homogeneous second-order differential equation with exponential forcing. The method of solving will involve finding the complementary (homogeneous) solution and the particular solution, using the initial conditions to find the constants.

Would you like a detailed solution for one of these questions, or for all? Let me know how you'd like to proceed!

Here are 5 related questions to consider:

1. How do you find the complementary solution of a non-homogeneous differential equation?
2. What is the method of undetermined coefficients and when do you use it?
3. How can initial conditions help in solving second-order differential equations?
4. What are the characteristic roots and how do they affect the general solution?
5. How can trigonometric identities simplify non-homogeneous terms like $\cos^2 x$?

Tip: Always simplify non-homogeneous terms using appropriate identities (e.g., $\cos^2 x$ can be written as $\frac{1 + \cos(2x)}{2}$).