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The given problem asks to apply the method of **reduction of order** to find a second solution \( y_2 \) of a differential equation when one solution \( y_1(x) = 5x \) is already provided. The equation for \( w \) involves using \( y_1 \) and the given form of the differential equation:

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

We are asked to find \( w(t) \), where \( w(1) = -4 \). Let's break down the solution step-by-step.

### Step 1: Understanding the form of the equation
Given that \( y_1(x) = 5x \), its derivative is \( y_1'(x) = 5 \).

Now, the equation to solve is:

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

Substitute \( y_1 = 5x \) and \( y_1' = 5 \) into this:

\[
5x w' + (2(5) + p(x)(5x)) w = 0
\]

This simplifies to:

\[
5x w' + (10 + 5x p(x)) w = 0
\]

### Step 2: Solve for \( w(x) \)
Now, the equation is in the form of a first-order linear differential equation for \( w \). To proceed, we need the explicit form of \( p(x) \), which isn't provided in the image. In general, this can be solved using an integrating factor or other appropriate methods.

Given the boundary condition \( w(1) = -4 \), the solution would follow from solving the differential equation explicitly for \( w(x) \).

Let me know if you want further details about solving the equation, or if you'd like clarification on any specific part of this process!

Here are 5 relative questions that build on this problem:
1. How do you compute the integrating factor for first-order linear differential equations?
2. What is the general form of the reduction of order method?
3. How can we solve differential equations when the function \( p(x) \) is unknown?
4. What are the properties of a fundamental solution set for a second-order differential equation?
5. What is the role of boundary conditions like \( w(1) = -4 \) in solving differential equations?

**Tip**: When solving first-order linear differential equations, the integrating factor is a key tool. It is typically of the form \( e^{\int P(x) dx} \), where \( P(x) \) is the coefficient of \( w \).

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 y1 w' + (2y1' + p(x)y1)w = 0 so that w(1) = -4.

Learn how to apply the reduction of order method to find another solution y2, starting from y1(x) = 5x, in this differential equation problem. This solution includes steps to find w(t) where w(1) = -4 and demonstrates key concepts in solving second-order linear differential equations.

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