Math Help - Question with Diff Eq

1. Question with Diff Eq

I haven't had Diff Eq, and in my notes for an unrelated class, part of a way to solve a solution was with the following diff eq.

(d^2)x/d(t^2) + a^2 * x = 0 | a=constant; t=0; dx/dt = 0

I really don't know what to do.

Any help is greatly appreciated.

Thanks

2. Originally Posted by Katzenjammer
I haven't had Diff Eq, and in my notes for an unrelated class, part of a way to solve a solution was with the following diff eq.

(d^2)x/d(t^2) + a^2 * x = 0 | a=constant; t=0; dx/dt = 0

I really don't know what to do.

Any help is greatly appreciated.

Thanks
Supposing a solution of the form $x=e^{rt}$, we see that our auxiliary equation becomes $r^2+a=0$.

However, I ask this question: is $a$ a positive or negative constant?

I don't want to continue on until this is clarified.

3. Originally Posted by Chris L T521
Supposing a solution of the form $x=e^{rt}$, we see that our auxiliary equation becomes $r^2+a=0$.

However, I ask this question: is $a$ a positive or negative constant?

I don't want to continue on until this is clarified.
I have no idea, it was at the end of class and my Dynamics teacher said "finish this for next class."

I'm assuming it's a positive, since it's acceleration.

4. Originally Posted by Katzenjammer
I have no idea, it was at the end of class and my Dynamics teacher said "finish this for next class."

I'm assuming it's a positive, since it's acceleration.
Originally Posted by Chris L T521
Supposing a solution of the form $x=e^{rt}$, we see that our auxiliary equation becomes $r^2+a=0$.

However, I ask this question: is $a$ a positive or negative constant?

I don't want to continue on until this is clarified.
Ok, continuing on...

The solution to the characteristic equation is $r^2=-a\implies r=\pm i\sqrt{a}$.

So it follows that the general solution to the equation is $x(t)=c_1\cos\!\left(\sqrt{a}t\right)+c_2\sin\!\lef t(\sqrt{a}t\right)$

Are you sure there isn't another initial condition? Since its a second order equation, there should be one more initial condition.

5. Just peeked back my notes, he only gave the three initial conditions.

It wouldn't surprise me if he left something out, he's known to do that. :-p