Chapter 8 - Linear Differential Equations of Order n - 8.5 Oscillations of a Mechanical System - Problems - Page 539: 1

See below

Given: $\frac{d^2y}{dt^2}+4y=0$ Yields: $\frac{d^2y}{dt^2}+w_0^2y=0$ where $w_0=\sqrt 4=2$ The time interval: $T=\frac{2 \pi}{w_0}=\pi$ The complementary function for the given equation is: $y(t)=c_1\sin 2t+c_2\cos 2t$ Since $y(0)=2,y'(0)=4$ We have $c_2=2,c_1=2$ Hence, the general solution is $2\sin 2t+2\cos 2t=2\sqrt 2\cos(2t+\frac{\pi}{4})$ The amplitude is $A=2\sqrt 2$ Phase $\phi=\frac{\pi}{4}$