I Solutions to Simple Harmonic Motion second order differential equation

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The second order differential equation for simple harmonic motion is expressed as d²s/dt² = -k²s, where k is a positive constant. The primary solution is sinusoidal, represented by s = A cos(ωt + δ), with A as amplitude, ω related to frequency, and δ as phase displacement. There are indications that other functions may also satisfy this equation, suggesting the existence of additional independent solutions. The discussion emphasizes that the amplitude and phase parameters encompass the two independent solutions typically found in such equations. Exploring these alternative solutions could provide a deeper understanding of simple harmonic motion.
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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A second order differential equation has two independent solutions. With the amplitude and phase parameters you are covering both of those.
 

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