# Homework Help: 2nd order differential equation: undetermined coefficients

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1. Dec 6, 2014

### dmoney123

1. The problem statement, all variables and given/known data

y''-y=t-4e^(-t)

2. Relevant equations

method of undetermined coefficients

3. The attempt at a solution

solving for characteristic equation first

y''-y=0

r^2-1=0

c_1e^(-t)+c_2e^(t)

RHS

particular solution

t-4e^(-t)

y_p(t)= At+B+Ce^(-t)

y_pt'(t)=A-Ce^(-t)

y_p''(t)=Ce^(-t)

plug into LHS

Ce^(-t)-At-B-Ce^(-t)=t-4e^(-t)

-A=t
A=-1

B=0

C cancels on the left.... Im not sure how to any further?

Thanks

2. Dec 6, 2014

### Staff: Mentor

No, this won't work, since e-t is already a solution in the complementary equation. Do you know what to do when you run into this?

3. Dec 6, 2014

### dmoney123

so the guess becomes At+B+Cte^(-t) right? When I plug that back in, it still cancels out

4. Dec 6, 2014

### Staff: Mentor

I'm not sure what you mean.
If yp = At + B + Cte-t, then yp'' - yp has to equal t - 4e-t. Take yp and its second derivative, and substitute them into your nonhomogeneous equation to get A, B, and C.

5. Dec 6, 2014

### dmoney123

Got it... I was making a mistake with my derivative... I feel stupid.

Thanks Mark44!