Differential Equations - Variation of Parameters problem

[Nubcakes]

As the name suggest, this problem is an undetermined coefficients problems where variation of parameters is necessary to solve. As with my previous question; This is not a homework problem, but it is out of the textbook so I figured this would be the appropriate place to ask if I am doing it correctly.

Here is the initial problem where one is asked to find the general solution;

I know there are at least 4 different approaches to this problem, but nearly all will not work well on other similar problems. However, from what I can understand the "Variation of Parameters" technique can be used to solve almost any differential Equation problem in this format so long as you do not encounter an impossible-to-solve integral EXA:{ln(ln(x))}

That being said; here is my attempt at the problem using "Variation of Parameters";

Simple question; Did I do it right? If not where did I screw up?

 

[HallsofIvy]

As the name suggest, this problem is an undetermined coefficients problems where variation of parameters is necessary to solve.

No, "undetermined coefficients" and "variation of parameters" are two completely different methods of find a specific solution to a non-homogeneous linear equation.

In your equation for v'₂ you seem to have neglected a factor of cos(x) on the right. You have cos²(x) from the differential equation. To solve for v₂' you have to multiply by another cos(x). You should have v₂'= cos³(x).

 

[Nubcakes]

Whoops... I should have caught that that problem with V2. Anyway thanks for the help, got to remember to keep a close eye on everything in these problems... its pretty easy to loose track of parts of the problem since there are so many parts.