Question with Diff Eq

**Katzenjammer** · September 21st 2009, 04:56 PM

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

**Chris L T521** · September 21st 2009, 06:25 PM

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.

**Katzenjammer** · September 21st 2009, 07:00 PM

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.

**Chris L T521** · September 21st 2009, 07:07 PM

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.

**Katzenjammer** · September 21st 2009, 07:18 PM

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

Thanks for your help, though.

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