Solutions to Simple Harmonic Motion second order differential equation

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Trollfaz
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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
 
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