Second Order ODE - Variation of Parameters

[tracedinair]

Homework Statement

Find the general solution of the following diff. eqn.

y''(t) + 4y'(t) + 4y(t) = t^(-2)*e^(-2t) where t>0

Homework Equations

General soln - Φ_general(t) + Φ_particular(t)

Wronskian - Φ₁(t)Φ2₂'(t) - Φ₂(t)Φ₁'(t)

The Attempt at a Solution

I'm solving by variation of parameters.

First solving for the general solution, y'' + 4y' + 4y = 0

r² + 4r + 4 which factors into (r+2)(r+2), so r = -2, -2.

So the gen solution is y = c1₁e^(-2t) + c₂e^(-2t)

Now solving for the particular solution.

Φ₁ and Φ₂= e^(-2t)

The Wronskian here ends up being 2e^(-4t) - 2e^(-4t) which equals zero.

What went wrong here? I know the Wronskian cannot equal zero here. This is where I am stuck.

 

[tracedinair]

Wow, two seconds after I posted this I realize what I did wrong. Φ₂ is equal to te^(-2t). Not e^(-2t). But that still equals zero haha.

Edit again: I didn't use product rule. That was the problem.