Help solving non homogenous second order ODEs

[seang]

I'm using the method of undetermined coefficients here, but I'm either not making the correct ansatz or I'm just confused on the method.

The problem is 2y'' + 3y' + y = t^2.

I gussed Y = At^2. Is this correct? It doesn't solve the differential equation, which is the only check I know.

So from there (assuming Y is correct) I plug in and get 4A +6At + At^2 = t^2. Now I don't know what to do next.

 

[arildno]

Try a particular solution of the form Y=At^2+Bt+C where A,B,C are constants. You'll get 3 equations with 3 unknowns to solve.

 

[hypermorphism]

The method of undetermined coefficients relies on all linear combinations of the linearly independent derivatives of the RHS. You really don't know yet whether there are lower power terms on the LHS that have simply canceled out. So your Y should be Y(t) = At^2 + 2Bt + 2C = At^2 + Dt + E. Plug this Y into the original equation and you will get a system of 3 linear equations that is easily solved for the coefficients A, D and E.