Help with Method of Undetermined Coefficients.

I am having trouble solving this problem with the method of undetermined coefficients:

$(\frac{d^2}{dt^2}+2\frac{d}{dt}+5)y=12e^t-34sin(2t)$

Work: To solve the homogeneous equation, I first tried to find the roots of the auxiliary equation, which I believe is $r^2+2r+5=0$. However, I cannot get a clean root here and I am unsure of what to do in this case.

For $r^2+2r+5=0$, the roots are

$$r_{1,2} = -1 \pm ~ 2i.$$

This means that:

$$y_h(t)= e^{-t}(c_1 \cos 2t + c_2 \sin 2t)$$

Can you proceed with the particular? Hint, choose:

$$y_p = a e^t + b \cos 2t + c \sin 2t$$

Sub back into original ODE and solve for the constants.

You should get:

$$a = \dfrac{3}{2}, b = 8, c = -2$$

Write:

$$y(t) = y_h(t) + y_p(t)$$

• Hi thank you, I looked up in my textbook and managed to get $y_h$. I now understand how to solve the second part with the guess equation. However, I still do not understand how to come up with that guess in the first place. Is there some sort of rule? – mrQWERTY Mar 24 '14 at 2:57
• That table was extremely helpful. Thank you. – mrQWERTY Mar 24 '14 at 3:00
• @Froggy: By the way, see the nice Table in Section 2.6 of math.msu.edu/~gnagy/teaching/ode.pdf . Regards – Amzoti Mar 25 '14 at 18:09