Solving ODE with Fourier Transform: $u(x) - u''(x) = x^2$

Question

Solve:

$$u(x) - u''(x) = x^2$$

You can use: $$ \mathcal{F} \{ e^{-a|x|} \} = \frac{2a}{a^2 + s^2} $$

I am new to Fourier transforms. I understand that limits have to be used but don't know how to start. Any help will be appreciated.

Answer 1

You probably are to use that $$ x^2=\left.\frac{\partial^2}{\partial a^2}e^{-a|x|}\right|_{a=0} $$ and consider first the solution of $$ u_a(x)-u_a''(x)=e^{-a|x|} $$ to then compute $$ u=\left.\frac{\partial^2}{\partial a^2}u_a(x)\right|_{a=0}. $$

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Check against the solution you get via undetermined coefficients, $u_p=ax^2+bx+c$.