- #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!