1
$\begingroup$

I have the following integrand:

int = Sin[Sqrt[-g^2 + omega^2]*(t - tp)]*Exp[-g*(t - tp)]*A*Exp[-(tp - t0)^2/sigma^2]*Cos[Omega*tp]/Sqrt[-g^2 + omega^2]

and am trying to integrate it with:

Integrate[int, {tp, 0, t}, Assumptions -> {g > 0, omega > 0, Omega > 0, A > 0, sigma > 0, t > 0, t0>0}]

but Mathematica is not able to do it (I have tried to do this with no assumptions as well). Maple does the integration in under a second and returns a solution (which can be expressed in terms of the Erf functions).

Is there a way to help Mathematica calculate this integral?

... and just to note, the result of the integration is a solution of damped harmonic oscillator, driven by a force:

A*Exp[-(t - t0)^2/sigma^2]*Cos[Omega*t]
$\endgroup$
1
  • $\begingroup$ the small omega is the natural frequency, while the large omega (i.e. Omega) is the frequency of the drive. $\endgroup$ – user2562235 Jul 21 '13 at 23:16
5
$\begingroup$

This takes about 30 seconds to evaluate on my computer:

 Integrate[int // TrigToExp // Expand, {tp, 0, t}, 
  Assumptions -> {omega > g > Omega > 0, A > 0, sigma > 0, t > 0, t0 > 0}]

Also if it is a solution of a linear ODE (with constant coefficients?) then perhaps it can be found directly with DSolve.

$\endgroup$
1
  • $\begingroup$ thanks for taking a look at this. Doing TrigToExp, and Expanding does indeed work! I should note that you added extra assumptions in your answer, but going back to the assumptions that I specified in my question also works. $\endgroup$ – user2562235 Jul 21 '13 at 21:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.