1. ## first order DE

Which method do I use to solve

(1 + t^2) dy/dt + 2ty = -picos(pi*t) y(0) = 0

I couldn't find a method to match this DE

2. Well I don't if you can see it or not, but the LHS says this

ay'+a'y = ay (product rule)

since d/dt(1+t^2) = 2t and y->y'

we can say LHS = (1+t^2)*y = -pic0s(pi*t)

then y=-picos(pi*t)/1+t^2

or you can simply use the p(x)=e^int(something)

3. I think you actually mean

$ay' + a'y = \frac{d}{dt}(ay)$.

So that means

$\frac{d}{dt}[(1 + t^2)y] = -\pi \cos{(\pi t)}$.

Go from here.