The image contains a set of questions based on a 2nd-order homogeneous ordinary differential equation (ODE). Here is a breakdown of the tasks:

Given Information:

The ODE is: $t^2 \frac{d^2x}{dt^2} + 2t \frac{dx}{dt} - 2x = 0$ with boundary conditions: $x(1) = 4, \quad x(2) = 6.25$ Numerical methods must use a step size $h = 0.2$.

Part (a):

Convert the ODE to a system of 1st-order ODEs.

You are asked to rewrite the 2nd-order ODE as a system of 1st-order ODEs. The system of equations is provided: $\frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} x_2 \\ \frac{2x_1}{t^2} - \frac{2x_2}{t} \end{bmatrix}$ where $x_1 = x$ and $x_2 = \frac{dx}{dt}$.

Part (b):

Perform 1 iteration of the RK2 method by hand (using initial conditions from part (a) with $x'(1) = 2$) and calculate the $x(2)$ vector. Numerical results should be displayed with 6 decimal places.

Tasks:

• Hand calculation for 1 iteration using RK2.
• Use the Maple program to find $x(2)$.
• Ensure no rounding errors in the calculations.

Part (c):

Perform 1 iteration of the RK4 method by hand (using $x'(1) = 3$) and find $x(2)$ similarly as in part (b), but now with the RK4 method.

Tasks:

• Hand calculation for 1 iteration using RK4.
• Use the Maple program to find $x(2)$.
• Display all results to 6 decimal places.

Part (d):

Use the shooting method to estimate $x'(1)$ to 4 decimal places, using the results from parts (b) and (c).

Do you want to go over the RK2 and RK4 methods in detail, or any specific part of the process?

Here are 5 related questions for further exploration:

1. How does the shooting method handle boundary value problems in ODEs?
2. What are the key differences between RK2 and RK4 methods in terms of accuracy?
3. How is the Maple software helpful in solving systems of ODEs?
4. Why is it important to avoid rounding errors in numerical methods?
5. What are the typical applications of converting higher-order ODEs to a system of first-order equations?

Tip: When performing manual calculations for RK2 or RK4, break down the process step by step to avoid common mistakes, especially during intermediate calculations.