Differential Equation (see comments)

Question

unimatix · Answer

Find the values of w for which cos(wt) satisfies. 

$$\frac{ d^2 }{ dt^2 } + 49y = 0$$

stbest · Answer

i wouldve helped but i have no clue how to do this

arthur326 · Answer

Please clarify the equation. You have an operator, $\dfrac{d^2}{dt^2}$, by itself. That doesn't make sense. The second derivative with respect to $t$ of what?

unimatix · Answer

Thanks typo. sorry.

unimatix · Answer

It should be $$\frac{ d^2y }{ dt^2 }$$

arthur326 · Answer

Okay. What's a reasonable thing to do to get started?

arthur326 · Answer

(Or are you stuck?)

unimatix · Answer

Separate by variable and take the integral of each side?

arthur326 · Answer

Solving this differential equation (and proving you found all the solutions) takes some work, but what makes this problem easy is that they already give you a set of functions.

arthur326 · Answer

The problem is asking: for what values of $\omega$ does the function $y=\cos\omega t$ satisfy the given differential equation.

arthur326 · Answer

(I meant to put a question mark in the last post.)

unimatix · Answer

Ok.

arthur326 · Answer

Have you tried plugging in $y = \cos \omega t$? What do you get?

unimatix · Answer

$$\frac{ d^2 \cos(\omega t) }{ dt^2 } + 49\cos(\omega t) = 0$$

unimatix · Answer

Unsure of what to do next.

arthur326 · Answer

Problem solving tip I learned from the pros: If there's only one thing you can do, do that. We can compute $\dfrac{d^2}{dt^2} \cos\omega t $.

unimatix · Answer

$$\frac{ d }{ dt } \frac{ 1 }{ \omega }\sin(\omega t)$$

unimatix · Answer

Yes? No?

arthur326 · Answer

No, careful! I think you integrated instead of taking a derivative.

unimatix · Answer

So would I take the derivative twice here?

arthur326 · Answer

Yes. It's the second derivative with respect to $t$.

unimatix · Answer

$$-\omega \sin(\omega t)$$
$$-\omega^2 \cos(\omega t)$$

arthur326 · Answer

Good. What's the equation now?

unimatix · Answer

$$- \omega^2 cos(\omega t ) $$ = 0

arthur326 · Answer

You forgot a term.

unimatix · Answer

$$-\omega ^2 cos(\omega t) + 49y = 0 $$

arthur326 · Answer

Don't forget that we set $y=\cos \omega t$.

unimatix · Answer

Oh right!

unimatix · Answer

$$-\omega^2 cos(\omega t) + 49cos (\omega t) = 0$$

arthur326 · Answer

What can we do now?

unimatix · Answer

Solve for omega. So is omega just going to equal -7 and 7?

arthur326 · Answer

That's right!

unimatix · Answer

Awesome! Thank you so much!

arthur326 · Answer

You're welcome. :)