with ordinary differential equation $$(\frac{\hbar}{\gamma})^2\frac{d}{dt}c(t)+2i\delta(\frac{\hbar}{\gamma})^2\frac{d^2}{dt^2}c(t)+c(t)=0$$
I would like to try on Fourier transformation to solve for this 2nd order homogeneous linear ordinary differential equation.
So my thought process was, $$-2i\delta(\frac{\hbar}{\gamma})^2\to a_1 (complex number)$$ $$(\frac{\hbar}{\gamma})^2 \to a_2(complexnumber)$$ $$\frac{d}{dt}\to\boldsymbol A$$ and the eigenfunction of operator $\frac{d}{dt}$, the $\boldsymbol A$ $$(\int dw\exp(iwt)\to \boldsymbol V,\int dt\frac{1}{2\pi}\exp(-iwt)\to \boldsymbol V^{-1},$$ these are the matrix contains eigenstates, and the matrix contains eigen value in diagonal way is $$\boldsymbol D$$
Thus represents ODE as $$a_2\boldsymbol A^2c(t)+a_1\boldsymbol Ac(t)+c(t)=0$$ and $$a_2\boldsymbol V\boldsymbol D^2\boldsymbol V^{-1}+a_1\boldsymbol V \boldsymbol D \boldsymbol V^{-1} + \boldsymbol Vc(w)\boldsymbol V^{-1}=0 \iff \boldsymbol V[a_2\omega^2c(\omega)+a_1\omega c(\omega)+c(\omega)]=0$$
The thing that came up to my mind is that All differential equation that I had tried with fourier transformation was inhomogeneous or PDE. At this point I am lost what to do to get the solution.
Sould I solve for $\omega$? if then, what should I do next?