Part. integrals & solutions theory question

[Guest]

I have a homework question which says

Given:
A particular solution of y'' - y' - 2y = 20e^4t is f1 = 2e^4t
A particular integral of y'' - y' - 2y = 5 - 6t is f2 = 3t - 4

Find the general solution of y'' - y' - 2y = -5e^4t + 20 - 24t

The instructor says to refer to the theory in the book wich states:

Hypothesis.

1. Let f1 be a particular integral of
a0(x)y^n + a1(x)y^(n-1) + .... an-1(x)y' + an(x)y = F1(x)
2. Let f2 be a particular integral of
a0(x)y^n + a1(x)y^(n-1) + .... an-1(x)y' + an(x)y = F2(x)

Conclusion.

a0(x)y^n + a1(x)y^(n-1) + .... an-1(x)y' + an(x)y = k1F1(x) + k2F2(x)

I get lost trying to understand theory. Can someone please point me in the right direction?

 

[royhaas]

To find the general solution you still need to add in the solution to the homogeneous equation. Then you can use the theorem about superimposing the particular solutions.

The theorem just says that a linear combination of particular solutions to the same differential equation is also a solution to the differential equation composed by superimposing the outputs.