Question

Prove that the following two equations represent the simple harmonic motion-

X = A cosωt + B sinωt

X = A eiωt

Answer 1

1)

$$\frac{d^2x}{dt^2}=-\omega^2(A \cosωt + B \sinωt)=-\omega^2x\\\frac{d^2x}{dt^2}+\omega^2x=0$$

2)

$$\frac{d^2x}{dt^2}=-\omega^2(A e^{iωt} )=-\omega^2x\\\frac{d^2x}{dt^2}+\omega^2x=0$$