Converting Cos and Sin to Exponential in Differential Equations

[jesuslovesu]

Homework Statement

I am in the process of doing a physics problem with a differential equation that has the form:

y = Acos(kx) + Bsin(kx)

According to my notes, this can also be written as y =Ae^jkx + Be^-jkx, unfortunately I just don't see how to write the original equation like that.

Homework Equations

The Attempt at a Solution

I know that cos(x) = 1/2[ e^jx + e^-jx ]
sin(x) = 1/(2j) [ e^jx - e^-jx ]

I can almost see how you would get it for the cos(kx) term:
Since Real { cos(kx) + j sin(kx) } = e^jkx using Euler's identity.
But for sine, I am stumped.

 

[Ben Niehoff]

The A and B in the first equation are not the same as the A and B in the second equation. Give them all different letters. Then you can find the coefficients in the second equation in terms of the coefficients in the first (by means of a system of 2 linear equations).