Understanding Simple Harmonic Motion: Solving the Second Order ODE

[coverband]

The equation of simple harmonic motion is given as

(d2x)/(dt2)+xk^2=0

Could someone show me HOW we arrive at the general solution to this second order ODE

 

[Hootenanny]

The equation of simple harmonic motion is given as

(d2x)/(dt2)+xk^2=0

Could someone show me HOW we arrive at the general solution to this second order ODE

The solution is straightforward, simply take an Anzats of the form

$$x=Ae^{rt}$$

and substitute into the ODE. The value of the constants may be determined using appropriate initial conditions.