Can someone please help me with this? I have to find a solution using Wronskian and I'm having a very difficult time.

Question

Given y1, can you find y2? Find y1y'2 - y'1y2 = W(t).

Answer choices:

1. y2 = ty1

2. y2 = y1∫W(t) dt

3. y2 = y1 (c+∫(W(t)/y1^2) dt

4. y2 = y1∫(W(t)/y1) dt

Answer 1

Let consider y₁·y₂' - y₁'·y₂ = W(t) as linear equation for y₂; then if we divide this equation by y₁ we get:

y₂ - y₁'/y₁·y₂ = W(t)/y₁; Integrated factor μ = e ^(-∫y₁'/y₁dt = e^(-lny₁) = 1/y; Then multiplaying by this factor

_{you will get solution case 3..}