The motivation for the following is to convert the integro-differential equation


\begin{equation}
 \kappa\ddot x+\dot x=-kx+\beta\int\_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds,
\end{equation}


into a system of nonlinear ODEs


\begin{equation}
 \begin{split}
 \dot{x}&=v,\\
 \dot{v}&=\frac{1}{\kappa}(\beta M\_0-v-kx),\\
 \dot{M}\_n&=-M\_n+W^{(n+1)}(0)+vM\_{n+1}.
 \end{split}
 \end{equation}


Let


$$M\_n=\int\_{-\infty}^t W^{(n+1)}(x(t)-x(s))e^{s-t}ds,$$


for all $n\in\mathbb{N}$ where $W^{(n+1)}(z)$ is the $n+1$-th derivative of $W$ with respect to $z$. I can't see how the last line of the proof below is equivalent to $-M\_{n+1}+W^{(n+2)}(0)+vM\_{n+2}$? Perhaps I made the wrong substitution on line 3. Any help would be much appreciated.


Applying the Leibniz integral rule, observe that for $n=0$,
\begin{equation\*}
\begin{split}
 \dot{M\_0}&=\frac{\partial}{\partial t}\int\_{-\infty}^t W^{(1)}(x(t)-x(s))e^{s-t}ds\\
 &=W^{(1)}(0)+\int\_{-\infty}^t e^{s-t}\left(\dot{x}W^{(2)}(x(t)-x(s))-W^{(1)}(x(t)-x(s))\right)ds,\\
 &=-M\_0+W^{(1)}(0)+\dot{x}M\_1.
\end{split}
\end{equation\*}
The inductive step is then
\begin{equation\*}
\begin{split}
 \dot{M}\_{n+1}
 &= \frac{\partial}{\partial t} \left(\frac{\dot{M}\_n+M\_n-W^{(n+1)}(0)}{v}\right),\\
 &=\frac{v\frac{\partial}{\partial t}\left(\dot{M}\_n+M\_n-W^{(n+1)}(0)\right)-\dot{v}(\dot{M}\_n+M\_n-W^{(n+1)}(0))}{v^2},\\
 &=\frac{v\ddot{M}\_n+v\dot{M}\_n-\dot{v}\dot{M}\_n-\dot{v}M\_n-vW^{(n+2)}(0)+\dot{v}W^{(n+1)}(0)}{v^2},\\
 &=\frac{\ddot{M}\_n+\dot{M}\_n-\dot{v}M\_{n+1}-W^{(n+2)}(0)}{v}.\\
\end{split}
\end{equation\*}



Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$
Just as you did for $n=0$, for any $n\ge0$ write
\begin{equation\*}
\begin{split}
 \dot{M\_n}&=\frac{\partial}{\partial t}\int\_{-\infty}^t W^{(n+1)}(x(t)-x(s))e^{s-t}ds\\
 &=W^{(1)}(0)+\int\_{-\infty}^t e^{s-t}\left(\dot{x}(t)W^{(n+2)}(x(t)-x(s))-W^{(n+1)}(x(t)-x(s))\right)ds \\
 &=-M\_n+W^{(n+1)}(0)+\dot{x}M\_{n+1},
\end{split}
\end{equation\*}
to get the desired result.


Induction is not needed here. Also, your multiline differentiation display is incorrect. In particular, the derivative of the constant $W^{(n+1)}(0)$ is $0$, not $W^{(n+2)}(0)$.


