Fourier Transform of product of heaviside step function and another function

[Dextrine]

Homework Statement

A damped harmonic oscillator is driven by a force of the form f(t)=h(t) t^2 Exp(-t), where h(t) is a Heaviside step function. The Oscillator satisfies the equation x''+2x'+4x=f(t). Use pencil-and-paper methods involving Fourier transforms and inverse transforms to find the
response of the oscillator, x(t), assuming that x(0)=0 and x'(0)=1.

Homework Equations

The Fourier Transform F[f(t)]
The Inverse Fourier Transform F^(-1)[f(ω)]
Integration by parts

The Attempt at a Solution

First thing I did was take the Fourier transform of the left hand side, which I'm sure I got correct. The part I'm stuck on is taking the Fourier transform of f(t). I used integration by parts using
u=t^2/e^t dv=h(t)(Exp[i ω t])
du=(2t-t^2)/e^t v=πδ(ω)+i/ω

but now when I try to complete the integration, i get an integral that does not converge because of the i/ω. There have been multiple heaviside problems that I have been working on that all end up the same, and I'm not sure where to go from here.

 

[Dextrine]

In general actually, how would one go about solving the Fourier transform of a function with the form f(t)H(t)

for example, how could I do the Fourier transform of tH(t)? Again, I've tried doing it by parts, but nothing good comes of it

 

[vela]

The way you split the integrand up is unusual. The step function allows you to write the integral as
$$\int_{-\infty}^\infty t^2e^{-t}h(t) e^{i\omega t}\,dt = \int_0^\infty t^2e^{(i\omega-1)t}\,dt,$$ which you can integrate by parts. A trick you could use to avoid integration by parts is to note that $t^2 e^{i\omega t} = -\frac{\partial^2}{\partial \omega^2}e^{i\omega t}$, so you can say
$$\int_0^\infty t^2e^{(i\omega-1)t}\,dt = -\frac{\partial^2}{\partial \omega^2}\int_0^\infty e^{(i\omega-1)t}\,dt$$

 

[Dextrine]

Thank you so much you are a life saver. My homework became super easy after your post!

 

[paranormal]

It seems like you are on the right track with using integration by parts. However, instead of using the delta function in your calculation for v, try using the Fourier transform of the Heaviside step function, which is 1/(iω) + πδ(ω). This will give you a convergent integral and allow you to complete the calculation. Keep in mind that you will also need to use the convolution theorem to simplify the Fourier transform of the product of the Heaviside step function and the exponential function.

Once you have the Fourier transform of f(t), you can then use the inverse Fourier transform to find the solution for x(t). Remember to also use the initial conditions given in the problem to solve for the constants in your solution. Hope this helps!