https://file.aiquickdraw.com/mathbot/file/1728381059099h20598jn.jpg

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 \( \frac{1 + \cos(2x)}{2} \)).

Solve the following second-order differential equations:

1. (a) 2 d²y/dx² - 7 dy/dx - 4y = e³ˣ
(b) d²y/dx² + 4 dy/dx + 4y = 2 cos²x
(c) d²y/dx² + dy/dx - 2y = 2 cosh(2x)

2. Solve the equation d²x/dt² + 4 dx/dt + 3x = e⁻³ᵗ given the initial conditions x(0) = 1/2 and dx/dt(0) = -2.

This problem involves solving second-order non-homogeneous differential equations, using methods like undetermined coefficients and simplifying non-homogeneous terms like exponential, trigonometric, and hyperbolic functions. Detailed initial conditions provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation