Deriving the equation for simple harmonic motion.

Question

I am trying to derive the solution to the equation of simple harmonic motion without guessing the sin/cos result. I know I have seen this proof somewhere, but I can't find anything about it online. All I can find are sources using the guessing technique.

In a nutshell, I would be grateful if someone could point me in the direction of a proof that the most general solution to

$$\frac{d^2x}{dt^2} - k^2x =0$$

is

$$x(t)= A \cos(kt+\phi) = B\cos(kt) + C\sin(kt)$$

Answer 1

Hint: A good start should be writing the simple harmonic motion equation correctly

$$\frac{d^2}{dt^2} x(t) + \frac{k}{m} x(t) = 0$$

Thus writing $x(t) = e^{rt}$ you should find $r^2 + k/m = 0$ which implies $r = \pm i\sqrt{\frac{k}{m}}$. Now remember that $$e^{\mu it} = \cos (\mu t ) + i \sin (\mu t)$$ and that combinations of such are also solutions.

Answer 2

If $x'' + ax = 0$, multiplying by $x'$, $x'x'' + axx' = 0$, or $2x'x'' + 2axx' = 0$, or $(x'^2)' + a (x^2)' = 0 $.

Integrating, $x'^2+ax^2 = c $.

There a number of ways to solve this.