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How do we find the general solution of:

$mu$''+$ku$=0

This is the equation of motion with a damping coefficient of 0.

The characteristic equation is $m$r$^2$+$k$=0.

From here, how do we find the complex roots and get it to look like the following:

$u(t)$=$A$$cos$${w_0}$$t$+$B$$sin$${w_0}$$t$

Any help is greatly appreciated.

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This is the Harmonic Oscillator equation, possibly one of the most common in theoretical physics.

So lets put your equation in the form

$$ u'' + \frac{k}{m} u = 0 $$ and let

$$ \frac{k}{m} = \omega^2 $$ for ease.

As you assumedly did to get that correct characteristic equation, we take a general solution $$ u(t) = A\exp(rt) $$ giving you $$r = i\omega$$

Hence by ODE theory, a general solution is given by $$u(t) = A \cos(\omega t) + B \sin(\omega t)$$

as you correctly had.

Does this help?

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