Need some help please..
Obtain the general solution:
y''+y=12(cos x)^2

Question

Need some help please..
Obtain the general solution:
y''+y=12(cos x)^2

anonymous · Answer

Problem shows as:
$$y''+y=12\cos^2x$$
I found my Yc, but i need help finding Yp please

anonymous · Answer

$$Yc=C1cosx+C2sinx$$

jtvatsim · Answer

I'm thinking maybe a way to start is to integrate cos^2(x) twice... I'm still trying to work it out.

anonymous · Answer

I was wondering if maybe i need to use a trig identity..

jtvatsim · Answer

maybe we can start by recognizing that
$$\cos^2 x = \frac{1}{2}(1 + \cos(2x))$$ so the question is essentially dealing with $$\cos(2x)$$

jtvatsim · Answer

phew, I think I found it.

anonymous · Answer

ahhhhhh.... i see, ok so would Yp=A+B(cos2x)?

jtvatsim · Answer

that should probably work, I used more terms because I wasn't sure what I was going to need, but if you get something like Yp = -2 cos 2x + 6 as your final answer, then your choice worked.

jtvatsim · Answer

yeah, your choice works also (and is much faster btw). :)

anonymous · Answer

Awesome! Thanks for your help! @jtvatsim

jtvatsim · Answer

not a problem, you hit the nail on the head when you were thinking of trig identities... you deserve a medal yourself. :)

anonymous · Answer

Right on! My first one lol @jtvatsim

jtvatsim · Answer

It's always nice to get medals on your own questions... doesn't happen too often for me. :)