Second order ODE with dirac delta and fourier tranform I am struggling with the following equation
$$
\tau^2 \ddot{x}=x-m^2-Dm\tau^2 \delta(t)
$$
This equation can be easy solved by applying the Fourier transform on both sides
$$
(1+\omega \tau^2) \widehat{x} =m^2 \delta(\omega)+2D\tau
$$
which gives
$$
 \widehat{x} = \frac{m^2 \delta(\omega)+2 D\tau}{1+\omega \tau^2}
$$
and by applying the inverse transform one gets
$$
x(t) = m^2 + D e^{-|t|/\tau}
$$
One the other side one could consider the first equation and multiply both sides by $\dot{x}$ (using the same idea explained here Second-order nonlinear ODE with Dirac Delta ). After few passages one gets
$$
\frac{d}{dt} \left( \tau^2\frac{\dot{x}^2}{2} + V(x) \right) = - 2 D \delta(t) \dot{C}(0)
$$
where
$$
V(x) = -\frac{x^2}{2} + m^2 x
$$
Following the same procedure used here Second-order nonlinear ODE with Dirac Delta , one has
$$
\frac{d}{dt}[\tau^2\frac{\dot{x}^2}{2} + V(x)] = 0 \ \, t>0
$$
where
$$
\tau^2\frac{\dot{x}(0)^2}{2} + V(x(0)) = -2 D \dot{x}(0) 
$$
fixes the initial condition. One can realize that $m^2+K e^{-t/\tau}$ is a solution of the first order differential equation and $\tau^2\frac{\dot{x}^2}{2} + V(x) = m^4/2$ 
The problem is that if I try to fix $K$ using the last equation, I do not get $K=D$ as obtained applying the Fourier transform, but $\tau m^4 /(4 D)$. I do not understand where the second method fails and why they give different results.
 A: Substitutions
\begin{equation*}
s=t/\tau ,\;x(t)=m^{2}y(s),\;u(s)=y(s)-1,\;u(s)=\frac{D\tau }{m}v(s)
\end{equation*}
Then
\begin{equation*}
\partial _{s}^{2}v(s)=v(s)-\delta (s)
\end{equation*}
Fourier transform
\begin{equation*}
f(s)=\int d\omega \exp [-i\omega s]\tilde{f}(\omega ),\;\tilde{f}(\omega )=%
\frac{1}{2\pi }\int ds\exp [+i\omega s]f(t)
\end{equation*}
Now
\begin{eqnarray*}
v(s) &=&\frac{1}{2}\exp [-|s|] \\
\partial _{s}v(s) &=&\frac{1}{2}\{-\exp [-s]\theta (s)+\exp [+s]\theta (-s)\}
\\
\partial _{s}^{2}v(s) &=&v(s)-\delta (s)
\end{eqnarray*}
Note that, given the solution obtained by Fourier transforming,
\begin{equation*}
\{\partial _{s}v(s)\}\delta (s),\;\frac{1}{\partial _{s}v(s)}\delta (s)
\end{equation*}
are ill-defined. Thus the second method becomes problematic if the integral
over $[-\varepsilon ,+\varepsilon ]$ is contemplated. Of course you can
proceed assuming that everything is all right and maybe end up with an
answer but you cannot expect it to coincide with the solution already
obtained.
