Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is it possible to solve this differential equation: $$\ddot{x}(t)+\omega^2x(t)=k\delta(t)$$ where $k$ is a constant and $\delta(t)$ the Dirac delta function? Is it possible alternatively, to know something about the spectrum $X(\omega)$ of $x(t)$ without solving the equation? Thanks

share|cite|improve this question
up vote 1 down vote accepted

You can solve the equation using the Laplace transform. Wolfram Alpha has the details

share|cite|improve this answer

If for t<0 the oscillator is at rest, x=dx/dt=0, then the effect of this delta force is providing an initial velocity at t=0. By integrating the equation over an infinitesimal interval [-dt,dt] we find that at time t=dt the velocity is dx/dt=k and the displacement is still x=0. So these are the initial conditions at t=0, and the solution is x=(k/omega)*sin(omega*t) for t>0; x=0 for t<0.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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