General solution to given DE

Question

Which of the following is the general solution of some DE of the form:

y''' +by'' +cy' +dy=0

where b,c,d are real constants:

a) C1 + C2.t + C3(5t+1)

b) C1.cos(t) + C2.sin(t) + C3

c) C1.e^(3t).cos(2t) + C2.e^(3t).sin(2t)

Answer 1

The general solution of the DE as given is

y=C₁exp(r₁x) + C₂exp(r₂x) + C₃exp(r₃x)

if r₁, r₂ and r₃ are the roots of x³ + bx² + cx + d and the roots are real and distinct.

If one of the roots is zero and the other 2 are complex conjugates, b) will be a result.