Undergrad Solutions to Simple Harmonic Motion second order differential equation

[Trollfaz]

All simple harmonic motion must satisfy
$$\frac{d^2s}{dt^2}=-k^2s$$
for a positive value k.
The most well known solution is the sinusoidal one
$$ s=Acos/sin(\omega t + \delta)$$
A is amplitude, $\omega$is related to frequency and $\delta$ is phase displacement.
My lecturer said that there might be other functions that satisfy the second order differential equation and I would like to know some other solution to the equation

 

[Orodruin]

A second order differential equation has two independent solutions. With the amplitude and phase parameters you are covering both of those.