[Math] How to Solve the Coupled Differential Equations

Question

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED

$\ddot{y}=\omega \dot{z}$

$\ddot{z}=\omega (\frac{E}{B}-\dot{y})$

I am at a loss as to how I may approach the problem to solve. Any help is appreciated.

Answer 1

Let $q=\dot{y}$. Then

$$\ddot{q} = \omega \ddot{z} = \omega^2 \left ( \frac{E}{B}-q\right)$$

or

$$\ddot{q}+\omega^2 q = \frac{E}{B} \omega^2$$

Can you solve this?