Dif eq with particular solution

[UrbanXrisis]

find the particular solution for y''+25y=50sin(5t)

I am using variation of parameter:

$$y_p=U_1e^{5x}+U_2e^{-5x}$$
$$y_1=e^{5x},y_1'=5e^{5x},y_2=e^{-5x},y_2'=-5e^{-5x}$$
$$U_1=- \int \frac{e^{-5x}50sin(5t)}{-10}=-0.5e^{-5x} (cos5x+sin5x)$$
$$U_2= \int \frac{e^{5x}50sin(5t)}{-10}=0.5e^{5x} (cos5x-sin5x)$$

$$y_p=-0.5e^{-5x} (cos5x+sin5x) e^{5x}+ 0.5e^{5x} (cos5x-sin5x)e^{-5x}$$
$$y_p=-sin5x$$

I know this is wrong because the check doesn't equal 50sin(5t)

any ideas?

 

[arildno]

Why do you use the e^5x-stuff??
They aren't solutions to the homogenous problem.

 

[arildno]

No. r^2=-25. What is r therefore?

 

[UrbanXrisis]

yep, i finished the problem. had to use Eulers formula, never would have guessed! thanks!

 

[HallsofIvy]

Surely your textbook discusses the case when the characteristic roots are complex?