Question

Hi all.
I need help understanding something my book dosn't go over well. For example dy/dx=-4y is a given solution and I need to find the differential equation. I simply isolate y then integrate. Getting y=e^(-4x) which is correct. My question is for something like (d^2y)/(dx^2)=-16y how is this process different because I cant get to the answer by just doing the integral twice. The answer by the way is y=sin(4x). Please help explain this Im taking an online class and the book dosn't explain this?

thank!

Answer 1

You have

(D^2+16)y = 0
so the solutions are
e^(±4ix) or sin and cos

(sin 4x)" = -16sin(4x)
(cos 4x)" = -16cos(4x)

If the answer is just sin(4x) then there must be some other conditions

Answer 2

I cant really follow how you got your work...