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Solving Underdamped Equation of motion

I am trying to solve a differential equation. The equation is the equation
of motion for a spring and dashpot. The solution is textbook. As an
exercise I would like to obtain the same solution using Mathematica. The
equation is:
x''+ 2 z w x' + w^2 x = y''

where y'', the forcing function, is just a variable here.

When mathematica solves it. It gives me a solution in the form of e to
some exponent. The form I want it in is in aCosb + cSind. I used the
ExpToTrig function but it gives me the equation in the form of Cosh and
Sinh. In order to get the output to produce what I want, I have to tell
Mathematica z is less than one. This corresponds to an underdamped system.
If I enter a value of z <1 and then do the ExpToTrig function I get the
solution I want. However it is evaluated at z = to that value and does not
give me the general solution. Does anyone know how I can get it to produce
the general solution for z < 1. In other words to constrain it at that set
of values?

The command I enter is below:
z has been replaced with zeta, and with omega

DSolve[{X''[t]+2* \[Zeta] *\[Omega]*  X'[t]+\[Omega]^2*X[t]==-z},X[t],t]


Carl Liggio
CLiggio at

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