## Fun with Trigonometry

The homogeneous solution is

Using variation of parameters we get a nonhomogeneous solution of

Distributing and regrouping

Reversing the cosine angle difference formula

Using the initial conditions $x(0)=0$, $\dot{x}(0)=0$, we get $c_1=\frac{-25}{18}$ and $c_2=0$

So a specific solution is

Regrouping and applying a difference to product formula

so