Suppose that a fourth order differential equation has a solution y=-9e^(3x)xcos(x).
a. Find such differential equation, assuming that it is homogeneous and has constant coefficients.
b. Find the general solution to this differential equation. x is the independent variable.
The Attempt at a Solution
I tried to find the derivatives of y (y',y'',y''',y'''').
Then, I placed unknown constants (c1, c2, c3, c4, c5) on each order and made a matrix with each row having coefficients from values containing e^(3x)cosx, e^(3x)xcosx, e^(3x)sinx, and e^(3x)xsinx. I made the columns all have the same order (such as all values from y'' on one column). I also made c1=1 since otherwise the matrix becomes 4*5 and that cannot be solved for the unknown constants. However, this did not work out.
I would sincerely appreciate your help. I mean, I can do those functions that have y=e^rx and y=cos(x), but combination of these makes my head ache.