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We are given the differential equation:

\[
-40x^2 y'' - 16x(x-5)y' + 16(x-5)y = 0, \quad x > 0
\]

### Part (a):
The problem provides that \( y_1(x) = 5x \) is a known solution. We are asked to use the **reduction of order** method to find another solution \( y_2(x) \) such that \( y_1 \) and \( y_2 \) form a fundamental solution set.

### Reduction of Order Method:
We assume the second solution is of the form:

\[
y_2(x) = v(x) y_1(x) = v(x) \cdot 5x
\]

Substitute this into the differential equation. However, for the reduction of order method, we can simplify the process by using the following form of the equation involving the Wronskian \( w(x) \):

We are given the form:

\[
y_1 w' + (2 y_1' + p(x) y_1) w = 0
\]

with the initial condition \( w(1) = -4 \). Our task is to solve for \( w(x) \).

### Step-by-Step Approach:

1. **First derivative of \( y_1(x) \)**:
   \[
   y_1'(x) = \frac{d}{dx} (5x) = 5
   \]

2. **General form**: The Wronskian differential equation is given by:
   \[
   y_1 w' + (2 y_1' + p(x) y_1) w = 0
   \]
   We substitute \( y_1 = 5x \) and \( y_1' = 5 \) into the equation to simplify.

Let’s now solve the simplified equation and apply the condition \( w(1) = -4 \).

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Would you like me to proceed with solving the Wronskian equation in detail? Feel free to ask if you want me to focus on any particular step.

### Related Questions:

1. How does the reduction of order method work in general for second-order linear differential equations?
2. What is the significance of the Wronskian in determining the independence of solutions?
3. Can you explain how to solve for \( v(x) \) after finding \( w(x) \)?
4. How does the initial condition \( w(1) = -4 \) influence the solution for \( w(x) \)?
5. What would be the next steps after finding \( w(x) \) to fully determine \( y_2(x) \)?

### Tip:
When using the reduction of order method, the key idea is transforming the problem into solving a simpler first-order equation using the known solution.

Consider the differential equation: -40x^2y'' - 16x(x-5)y' + 16(x-5)y = 0, x > 0. Given that y1(x) = 5x is a solution, apply the reduction of order method to find another solution y2 for which y1 and y2 form a fundamental solution set. Starting with y1, solve for w in y1w' + (2y1' + p(x)y1)w = 0 so that w(1) = -4.

Explore the solution to the differential equation -40x^2y'' - 16x(x-5)y' + 16(x-5)y = 0 using the reduction of order method. Given y1(x) = 5x, learn how to find another solution y2 such that y1 and y2 form a fundamental solution set. This problem includes concepts like the Wronskian and linear independence, suitable for advanced high school or college-level students.

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