# Need the correct time dependent function

## Homework Statement

The equation of motion of a free electron in an electric field is

m((d^2)x)/(dt^2) = -eE (Eq. 1)

If x and E have the time dependence exp(iwt), where w=angular frequency, then

-(w^2)mx = -eE (Eq. 2)

What does the time dependent function whose second derivative equals -(w^2)x look like?

## The Attempt at a Solution

I know from my papers that the electromagnetic wave is given by

E=E0exp(iwt),

and I guess the equation I'm after looks similar. The second derivative of x*exp(iwt) would give me the correct answer, but x=x0*exp(iwt) doesn't seem to do, because equation 2 says x and not x0.

x=x0exp(iwt)

d2x/dt2=-w^2 x0 exp(iwt) = -w^2 x

x=x0exp(iwt)

d2x/dt2=-w^2 x0 exp(iwt) = -w^2 x
Ah, I forgot that the factor exp(iwt) will still be there after the derivation

Thanks!

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