Nonhomogeneous 2nd order dif question?

[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 don't match up.
is there another combination I could try?

 

[Mark44]

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 = 3e^2t.

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 {e^3t, e^-t}.

For a particular solution, you would ordinarily try a solution of the form y_p = Ae^3t, 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 y_p = Ate^3t, and solve for the value of A that works.

 

[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 can't see what I am doing wrong? is there an error I missed or just a method I haven't used yet?

 

[Mark44]

That makes us even. I misread the function on the right side of your original DE. I thought you had it as 3e^3t, but what you had originally was 3e^2t, 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 y_p = At²e^-t and solve for the constant A. In other words, with this function, calculate y_p'' - 2y_p' - 3y_p = -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) = c₁e^3t + c₂e^-t + Ate^-t (with a specific value in place of A).

 

[footballxpaul]

thats my problem everything that should cancel isn't 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?

 

[Mark44]

How about this: y_p = Ate^-t + Bt²e^-t? I confess I'm a little rusty on this.

 

[footballxpaul]

hey don't worry about it, thanks for the help. Ill try that I think that might work