- #1

RoboNerd

- 410

- 11

## Homework Statement

show that the general solution of the differential equation d^2/dt^2 + 2 *alpha * dr/dt + omega^2 * r = 0,

where alpha and w are constant and R is a function of time "t" is R = e^(-alpha * t) * [ C1*sin( sqrt(omega^2 - alpha^2) * t) + C2*cos( sqrt(omega^2 - alpha^2) * t)

IF alpha^2 - omega^2 < 0.I was struggling over how to solve this problem and I would be very grateful if someone have me a badly needed hint.

## Homework Equations

None come to mind as of now[/B]

## The Attempt at a Solution

I said let R = f(t) = e^(d * t).

I then had the expression when I plugged it into the differential equation

d^2 * e^(d * t) + 2 *alpha * d * e^(d * t) + omega^2 * e^(d * t) = 0

I canceled the e^(d * t) term and I got the characteristic equation:

d^2 + 2*alpha*d + omega^2 = 0

I then applied the quadratic formula to solve for d:

d = [ -2 *alpha +/- sqrt( (2*alpha)^2 - 4*omega^2 ) ] / 2.

I thus get d = -alpha +/- sqrt (alpha^2 - omega^2 )

Now I have a huge problem. If alpha^2 - omega^2 < 0, then my d is non-real and I can not proceed from here.

I tried substituting in the functions sin(d*t) and cos(d*t) instead of e^(d*t) into the differential equation and solving, but then I become stuck as I can not eliminate the plugged in functions at all from the general expression.

Could anyone please be kind enough to help?

Thanks, and have a great day!