Solve a differential equation by Fourier transform: $ y''+6y'+5y=\delta (t).e^{-t}$

I know Fourier transform of $\delta (t)$ and $H(t).e^{-t} $ but I can't determine Fourier transform of $\delta (t).e^{-t}$.

Could you give me some hints? Thanks for helping

  • $\begingroup$ What definition of the fourier transform are you using? $\endgroup$ – John Doe Dec 31 '17 at 14:25

Hint: $$\delta(x-x_{0})f(x)=\delta(x-x_{0})f(x_{0})$$

  • 1
    $\begingroup$ @qbert He says he knows the transform of the delta function. He can use this to simplify $$\delta(t)e^{-t}=\:?$$ $\endgroup$ – eranreches Dec 31 '17 at 14:30
  • $\begingroup$ @qbert this says that $\delta(t)e^{-t}=\delta(t)$, so $\mathcal F[\delta(t)e^{-t}]=\mathcal F[\delta(t)]$ $\endgroup$ – John Doe Dec 31 '17 at 14:32

You don't need to memorize Fourier transforms, just how to integrate delta! $$ \mathcal{F}(\delta(t)e^{-t})(\xi)= \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-2\pi i \xi t}e^{-t}\delta(t)\mathrm dt\\ =\frac{1}{\sqrt{2\pi}}e^{-2\pi i \xi t}e^{-t}\vert_{t=0}=\frac{1}{\sqrt{2\pi}} $$


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.