# Second Order ODE

I'm not sure exactly how to solve this ODE. (dx^2)/(dt^2) + (w^2)x = Fsinwt, where x(0) = 0 and X'(0) = 0.
What I've got so far is:
x'' + w^2x = Fsinwt --> x(homogenous) = Acoswt + Bsinwt

I know I have to find a particular solution but I keep getting zero as a result which I know won't solve the ODE.

Also, I know that the answer is (F/2w^2)sinwt - (F/2w)tcoswt

## Answers and Replies

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vela
Staff Emeritus
Homework Helper
Show us how you solved for the particular solution.

For the particular solution I set x = Ccoswt + Dsinwt where C and D are arbitrary constants, so x' = -wCsinwt + wDcoswt, x'' = -w^2Ccoswt - w^2Dsinwt. so for the original equation
x'' + w^2x = Fsinwt, I have (-w^2Ccoswt - w^2Dsinwt) + w^2(Ccoswt + Dsinwt) = Fsinwt
but that reduces to 0 = Fsinwt which doesn't tell me anything about the particular solution because I have no values for C and D. I tried making x = tCcoswt + tDsinwt among others and even when I find some value for the particular solution, once I combine it with the homogeneous and try to solve for A and B I run into the same problem.

vela
Staff Emeritus