# Homework Help: Nonhomogeneous 2nd order dif question?

1. Jul 16, 2009

### footballxpaul

y''-2y''-3y=3e^2t find the general solution

I have tried Ate^t, Ate^2, Ate^3
none have worked they all leave extra variables that dont match up.
is there another combination I could try?

2. Jul 16, 2009

### Staff: Mentor

You need to do this in two parts:
Find the solutions to the homogeneous equation y'' - 2y' -3y = 0.
Find a particular solution to the nonhomogeneous equation y'' - 2y' -3y = 3e2t.

Your general solution will be all solutions to the homogeneous equation plus the particular solution.

For your homogeneous equation, a basis for your solution set is {e3t, e-t}.

For a particular solution, you would ordinarily try a solution of the form yp = Ae3t, but that won't work in your nonhomogeneous equation, since this is a multiple of one of the solutions of the homogeneous equation. Instead, try yp = Ate3t, and solve for the value of A that works.

3. Jul 16, 2009

### footballxpaul

opps wrote the wrong right side down.
its y''-2y'-3y=-3te^-t

i got the e^3t and e^-t already, the right side is still tricky
I keep getting 6At(e^-t)-12A(t^2)(e^-t)=-3t(e^-t)... and that was using A(t^3)(e^-t)
or
2A(e^-t)-6At(e^-t)=-3t(e^-t) with using A(t^2)(e^-t)
or
-2A(e^t)=-3t(e^-t) using At(e^-t)

I cant see what Im doing wrong? is there an error I missed or just a method I havent used yet?

4. Jul 16, 2009

### Staff: Mentor

That makes us even. I misread the function on the right side of your original DE. I thought you had it as 3e3t, but what you had originally was 3e2t, and you have changed that now to -3te-t.

What I said about the solution to the homogeneous equation is still valid. For your particular solution, try yp = At2e-t and solve for the constant A. In other words, with this function, calculate yp'' - 2yp' - 3yp = -3te-t. Group all of your terms by their power of t: t0, t, and t2. The coefficient of the t0 terms has to be zero, as does the coefficient of the t2 terms. The coefficient of the t term has to be -3.

Your general solution will be y(t) = c1e3t + c2e-t + Ate-t (with a specific value in place of A).

5. Jul 16, 2009

### footballxpaul

thats my problem everything that should cancel isnt canceling.

2A(e^-t)-6At(e^-t)=-3t(e^-t) with using A(t^2)(e^-t)
is what I am left with. What do I do with the 2A?

6. Jul 16, 2009