Missing step involving e for maxwell equ. solution

[d3nat]

Homework Statement

I am following along in my book (Laser Fundamentals by Silfvast, pg 14), and I cannot understand how they got from one step to the next.

Homework Equations

Since it's quite lengthy, I'm going to dive into the middle of it. If you need more background information, let me know

The Attempt at a Solution

( d^2*A_z ) / dz^2 + (w^2*A_z) / v^2 = 0
and
( d^2*A_t ) / dt^2 + (w^2*A_t) = 0

and then the book says that these are familiar forms and have the following solutions

A_z = C_1 e^i(w/v)z + C_2 e^-i(w/v)z
A_t = D_1 e^iwt + D_2 e^-iwt

I have following everything up to that point. I don't understand where they are getting the exponential part. I can somewhat see how it relates since if you look at the A_z part, the w^2/v^2 looks similar to how it goes to e^i(w/v)z
But I don't understand where the author pulled all of this information from. What are the missing steps from the top two lines to get to the bottom two lines.

Any advice is greatly appreciated.

 

[Dick]

They are assuming you know how to solve basic differential equations. There are two independent solutions of A''+c^2*A=0 (where the prime is differentiation with respect to the independent variable, call it t), A(t)=exp(ict) and A(t)=exp(-ict). Substitute them into check. Given that, a general solution is a linear combination of those two. Does that make it look more familiar?