Chapter 6.5, Problem 21P

Consider an oscillator satisfying the initial value problem

$u\text{'}\text{'}+{\omega }^{2}u=0,u\left(0\right)=0,u\text{'}\left(0\right)=v_0$.         (i)

(a) Let $x_1=u,x_2=u\text{'}$, and transform Eqs.(i) into the form

$\text{x'=Ax,}\text{x}\left(0\right)={\text{x}}_{\text{0}}$         (ii)

(b) By using the series(17), show that

$e^{\text{At}}={\text{I}}_{\text{2}}\cos \omega t+\text{A}\frac{\sin \omega t}{\omega }$.         (iii)

(c) Find the solution of the initial value problem (ii).