Answer
See below
Work Step by Step
Given: $\frac{d^2y}{dt^2}+w_0^2y=0$
The time interval: $T=\frac{2 \pi}{w_0}=\pi$
The complementary function for the given equation is:
$y(t)=c_1\sin w_0t+c_2\cos w_0t$
Since $y(0)=y_0,y'(0)=v_0$
We have $c_2=y_0,c_1=\frac{v_0}{w_0}$
Hence, the general solution is $y(t)=\frac{v_0}{w_0}\sin w_0t+y_0\cos w_0t=\sqrt (\frac{v_0}{w_0})^2+y_0^2\cos(wt+\tan^{-1}\frac{v_0}{y_0w_0})$
The amplitude is $A=\sqrt (\frac{v_0}{w_0})^2+y_0^2$
Phase $\phi=tan^{-1}(\frac{v_0}{y_0w_0})$
