Overdamped oscillator solution as hyperbolic function?

[Vitani11]

Homework Statement

Here is the equation for the general solution of an overdamped harmonic oscillator:
x(t) = e^-βt(C₁e^ωt+C₂e^-ωt)

Homework Equations

β decay constant
C₁, C₂ constants
ω frequency
t time

The Attempt at a Solution

I know (e^ωt+e^-ωt)/2 = coshωt and (e^ωt-e^-ωt)/2 = sinhωt but how do I implement this if there are constants? I tried to solve for the constants, but I just get a nasty expression in terms of x(o) and v(0) for each C (where v is the velocity) and this doesn't help with rewriting the functions.

 

[ehild]

Homework Statement

Here is the equation for the general solution of an overdamped harmonic oscillator:
x(t) = e^-βt(C₁e^ωt+C₂e^-ωt)

Homework Equations

β decay constant
C₁, C₂ constants
ω frequency
t time

The Attempt at a Solution

I know (e^ωt+e^-ωt)/2 = coshωt and (e^ωt-e^-ωt)/2 = sinhωt but how do I implement this if there are constants? I tried to solve for the constants, but I just get a nasty expression in terms of x(o) and v(0) for each C (where v is the velocity) and this doesn't help with rewriting the functions.

Write e^ωt and e^-ωt in terms of the hyperbolic functions cosh(ωt) and sinh(ωt).

 

[Vitani11]

Yes I know that is the goal but how do I do it if there are two constants that are different in front of each e term?

 

[Vitani11]

Nevermind I did it

 

[Vitani11]

Thank you