# Simple Harmonic Motion Differential Equation

## Homework Statement

A particle of mass m moves in one dimension under the action of a force given by -kx where x is the displacement of the body at time t, and k is a positive constant. Using F=ma write down a differential equation for x, and verify that its solution is x=Acos($$\omega$$t+$$\phi$$), where $$\omega$$2=k/m (omega squared, that is). If the body starts from rest at the point x=A at time t=0, find an expression for x at later times.

## The Attempt at a Solution

I think the differential equation they're looking for is,
a=-kx/m

As a=d2x/dt2

But from here I can't see where to go; integration of course leads to the wrong formula.

Kurdt
Staff Emeritus
Gold Member
You have the differential equation so just substitute the solution in and show that both sides are equal.

You have the differential equation so just substitute the solution in and show that both sides are equal.

So differentiate the solution given to us to get it in terms of acceleration and then just compare that with the a=-kx/m?

Kurdt
Staff Emeritus
Gold Member
Thats correct.

I'm sorry I'm still sightly confused; I now have:

$$a=-A\omega^{2}cos(\omega t+\phi)=\frac{-kx}{m}$$

Which implies that $$x=Acos(\omega t+\phi)$$ but doesn't really show why? Is this what you intended or am I missing something?

Kurdt
Staff Emeritus