I am trying to use the following result

Theorem:A pde of the form $$\frac{\partial w}{\partial t} = F\{x, \frac{\partial w}{\partial x},\frac{\partial^2 w}{\partial^2 x}\}$$ has an Additive separable solution $$w\left(x, t\right) = M t + N + \phi(x),$$ where $M$,$N$ are arbitrary constants, and the function $\phi(x)$, is determined by the ordinary differential equation $$F\{x, \frac{d \phi}{dx}, \frac{d^2 \phi}{d^2 x}\} = M.$$

The problem, I am studying has an initial condition and two Dirichlet boundary conditions, but I have had to perform a few change of variables to get the PDE, I am studying to look like the one in the above result. As a consequence of this the initial and the boundary conditions in my problem have transformed to the form \begin{eqnarray} \qquad w\left(f(x) + g(t), 0\right) &=& \delta\left(f(x) + g(t) - x_0\right),\\ w\left(f(x) + g(t) = 0, h(t)\right) &=& w\left(\pi / 2, h(t)\right) = 0 \qquad (t > 0) \end{eqnarray} where $f, g, h$ are some common but non-linear functions and $\delta$ is the dirac delta function, instead of having a form like $w\left(x, 0\right)$, $w\left(0, t\right)$ and $w\left(C, t\right)$, which I think is what is required to get the solution.

I am wondering what techniques or options are available for me to be able to use such initial and the boundary conditions. I have been searching to find such examples of problems, but I have not had much luck with this.

Any help will be greatly appreciated.

**Edit**

The pde I am trying to apply this theorem to, is \begin{equation} \frac{\partial w(x, t)}{\partial t} = - \frac{2 b \kappa x}{\left(x^2+1\right)^2} \frac{\partial w(x, t)}{\partial x} + \frac{b \kappa}{\left(x^2+1\right)^2} \frac{\partial ^2 w(x, t)}{\partial x^2}% \end{equation} and the initial condition is $$ w\left(x, 0\right) = \delta\left(x - x_0\right) $$ and the boundary conditions are $w = 0$ along the curves $$ \{(\pi/2, t): t> 0\} \text{ and } \{ (x,t) : \tan(x) + a (2\kappa t)^{\frac{1}{2\kappa}} = 0, t > 0\}, $$ and $a, b, \kappa$ are constants from my model. The ode corresponding to this problem is \begin{equation} \frac{d^2 \phi}{d^2 x} - 2 x \frac{d \phi}{dx} = \frac{M}{b \kappa}\left(x^2+1\right)^2 \end{equation} which has the general solution \begin{equation} \phi\left(x\right) = \frac{11 M x^2}{8 b \kappa}\, _2F_2\left(1,1;\frac{3}{2},2;x^2\right) - \frac{M x^4}{8 b \kappa} - \frac{7 M x^2}{8 b \kappa} + C_1 \int e^{x^2} \,dx + C_2, \end{equation} where $C_1$ and $C_2$ are constants of integration and we have a generalized hypergeometric function in the first term of the general ode solution.

boundary conditions(the boundary condition has to be somehow special as well). I think here, you want the boundary conditions on $w(x,t) - Mt-N$ to be independent of $t$. Just a guess, but perhaps you can generalize your ansatz to $w(x,t) = Mt + N + \phi(x) + \psi(x,t)$, where the purpose of $\psi(x,t)$ is precisely to get the boundary conditions into an acceptable form. $\endgroup$