Question

y^n+y = sin x + cos 2x
whats the general solution of this differential equation? thankssss

Answer 1

(dn(x, y))/(dx) = (cos(x)-2 sin(2 x))/(y^n log(y))

Answer 2

Do you mean the general solution to
\[{d^n y \over dx^n } + y = \sin x + \cos 2x\]
?

Answer 3

Ye

Answer 4

Is this a problem you made up? Even writing down the general homogeneous/complimentary solution is a bit of pain. I think finding a general form for all n of the particular solution is going quite tricky.

Answer 5

...going to be quite tricky

Answer 6

Or perhaps not. Let's look at an Ansatz/trial solution for yp in the form
yp = A sin x + B cos x + C sin 2x + D cos 2x

It's not too hard to write down the nth derivative of this yp, with two cases: n even, n odd; and then solve for A, B, C, D

Answer 7

The homogeneous part is likewise doable if we break it up into the cases where n is even and n is odd.