How do I prove the general solution for ODE with a real coefficient?

[simmonj7]

Homework Statement

If b =$$\sigma$$^2 > 0 (which implies b is real), then the general solution is Ce^(i$$\sigma$$x) + De^(-i$$\sigma$$x) = Esin($$\sigma$$x) + Fcos($$\sigma$$x) = Gcos($$\sigma$$x + H).

Homework Equations

N/A

The Attempt at a Solution

So I know how to get the first two forms of the solution. The first you just find the characteristic equation and then my book provides a theorem to just jump to the first solution. The second form of the solution, you just apply Euler's formula and then you will get to that form of the solution.

However, my problem is with the final form of the solution. I am not sure how to get there and what trig identities I would need to use (assuming I need to use some). Help please!

 

[Tangent87]

To show $$E\sin(\sigma x)+F\cos(\sigma x)=G\cos(\sigma x+H)$$ expand out the cos on the RHS and then compare coefficients, you will get equations which you can then solve for E and F.

But in general you can always write the sum of a sine and a cosine in terms of another sine or cosine with the same argument + a constant.