Could I get a bit of help with this?

[Zkirsche]

Ok, so I'm learning about ordinary differential equations for my interview at cambridge this week. (Which is why I'm inactive for mafia)

However, I was just stumped/confused by a question.

The question was, solve:

(d^2y/dx^2)+4(dy/dx)+4y = 2cos^2(x)

So I attempted to treat this as a normal linear inhomogeneous constant coefficient equation and began to use the method of variation of constants (or variation of parameters).

However, when I reduced this equation (by setting 2cos^2(x) to 0), and tried to find the two variables y₀ and y₁ which were solutions to the reduced equation.

However, the auxillary equation gives only one root. Meaning y₀ = y₁ = (A+Bx)e^(2x). Where A and B are arbitrary constants.

For the complementary equation, which usually comes in the form of y = A₀y₀ + A₁y₁, do I simply multiple out the brackets?

After doing so, do I simply replace A with V₁(x) and B with V₂(x) where V₁(x) and V₂(x) are functions of x and then work ahead normally, or is there a different route I need to take?

Any help would be much appreciated. :)

[Aleph]

I don't remember this crap very well, but isn't it supposed to be scaled or something? Like "2 and 2t" or whatever? Also why 2? Wouldn't it be -2?

lul I totally forgot DE. Also your syntax or whatever is way different from how we did it and not only do I not remember how we did inhomogeneous or that crap with matrices toward the end that made ungodly expressions look like common fucking sense, I almost don't even remember if that was a hazy dream or something that I actually went to school for...

[Zkirsche]

It would be -2 yes. I even wrote -2 so I don't know how that happened.