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## Homework Statement

Solve the following system for [tex]\mathbf{r}(t)[/tex]:

[tex]\frac{d^2\mathbf{r}}{dt^2}=-\frac{k}{m}\mathbf{r}.[/tex]

## Homework Equations

## The Attempt at a Solution

Now, I know how to solve for the magnitude of r (in fact, since it's the equation for the simple harmonic motion of a spring obeying hooke's law, I have it memorized), but I'd like to be able to solve for the component form. Here's what I've tried so far:

I start by guessing that [tex]\mathbf{r}[/tex] is in the form [tex]e^{\lambda t}\mathbf{u}[/tex], where [tex]\lambda[/tex] is an eigenvalue and [tex]\mathbf{u}[/tex] is an eigenvector. Plugging in, I have

[tex]\left(\lambda^2 + \frac{k}{m}\right) e^{\lambda t}\mathbf{u} = \matbf{0}.[/tex]

Since exp can never be 0, and it would be meaningless (I think) at this point to have u be 0, we can solve what's left for [tex]\lambda[/tex] and get

[tex]\lambda = \pm i \sqrt{k \over m}.[/tex]

I know that the answer should look something like[tex]\mathbf{r} = \mathbf{u}_1\cos{\sqrt{k \over m}}+\mathbf{u}_2\sin{\sqrt{k \over m}},[/tex]

but I'm not sure how to find the eigenvectors (u1 and u2) here. Any ideas?