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 $w$.