what do i set y= to to solve this?

y'' + 4y=sin2x

ive tried y=asin2x + bcos2x and not had much luck. help please

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- Feb 1st 2009, 02:36 PMsheep992nd order differential
what do i set y= to to solve this?

y'' + 4y=sin2x

ive tried y=asin2x + bcos2x and not had much luck. help please - Feb 1st 2009, 03:18 PMSoroban
Hello, sheep99!

Quote:

What do i set y= to to solve this?

. . . $\displaystyle y'' + 4y \:=\:\sin2x$

i've tried: $\displaystyle y \:=\:A\sin2x + B\cos2x$ . . . and not had much luck.

Of course not!

$\displaystyle y \:=\:A\cos2x + B\cos2x$ is a solution of the__homogeneous__equation.

You must try: .$\displaystyle y \;=\;Ax\cos2x + Bx\sin2x$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I came up with: .$\displaystyle y_p \;=\;-\tfrac{1}{4}x\cos2x$

The complete solution is: .$\displaystyle y \;=\;C_1\cos2x + C_2\sin2x - \tfrac{1}{4}x\cos2x$