Question

Suppose that a fourth order differential equation has a solution y = -9e^(4x)xsin(x) Find such a differential equation, assuming it is homogeneous and has constant coefficients.

Answer 1

ehhhhhh

-36 E^(4 x Sin[x]) (-4 Cos[x] + 48 Cos[x]^2 - 16 x^2 Cos[x]^2 +
192 x^2 Cos[x]^3 + 64 x^4 Cos[x]^4 + x Sin[x] -
112 x Cos[x] Sin[x] + 384 x Cos[x]^2 Sin[x] -
96 x^3 Cos[x]^2 Sin[x] + 256 x^3 Cos[x]^3 Sin[x] - 48 Sin[x]^2 +
12 x^2 Sin[x]^2 + 192 Cos[x] Sin[x]^2 - 192 x^2 Cos[x] Sin[x]^2 +
384 x^2 Cos[x]^2 Sin[x]^2 - 96 x Sin[x]^3 +
256 x Cos[x] Sin[x]^3 + 64 Sin[x]^4) ?

Answer 2

Take the first four derivatives and rewrite them in terms of the previous derivatives.