Could anyone please explain me how to solve differential equations using Fourier transform, distributions and Schwarz's spaces? I would be so grateful if you could help with a practical example -where i is the imaginary unit, i.e. $\sqrt{-1}$ (in theory I think I understood how it works but cannot actually solve any) \begin{equation} \frac{d^2u}{dx^2}+2i\frac{du}{dx}+4u=0 \end{equation} I tried to understand the passages but get stuck after applying the Fourier transform and transforming derivatives into multiplication by polynomials; in the example above, I get: \begin{equation} \left(-k^2-2k+4 \right) \hat{u}=0 \end{equation}
At this point, I suppose I should think about a linear combination of $\delta$ and derivatives: $ \sum_{n}c_n\delta^{\left(n\right)}$, since I have a distributional equation supported in zero. But I am not sure how to compute the coefficients and whether the method is right.
Factorize your polynomial $$-k^2-2k+4=-(k-a)(k-b)$$ Take $\phi \in C^\infty_c(-r,r),\phi(0)=1$ where $r<|b-a|/2$. For any $\varphi\in C^\infty_c(\Bbb{R})$ then $$\Psi(k)=\frac{\varphi-\varphi(a)\phi(k-a)-\varphi(a)\phi(k-a)}{-(k-a)(k-b)} \in C^\infty_c(\Bbb{R})$$
Thus $$\langle \hat{u},\varphi-\varphi(a)\phi(k-a)-\varphi(a)\phi(k-a)\rangle=\langle -(k-a)(k-b)\hat{u},\Psi\rangle = 0$$ and hence $$\langle \hat{u},\varphi\rangle = \varphi(a) \langle \hat{u},\phi(k-a)\rangle+\varphi(b) \langle \hat{u},\phi(k-b)\rangle$$ ie. $$\hat{u} = A\delta(k-a)+B\delta(k-b)$$ $$ u =\frac{ A e^{ita}+Be^{itb}}{2\pi}$$