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I'm working on a dynamic programing problem and I'm facing a Partial Differential Equation I'm struggling with. Unfortunatelly, I'm not a specialist on PDE's so any help would be very welcomed. Setting boundary/terminal conditions aside, for $V\equiv V(x,t):\mathbb{R}^2\to\mathbb{R}$ and $\sigma^2,m\in\mathbb{R}$, the PDE is the following:

$$V_t+\frac{1}{2}\bigg(\frac{\sigma^2 m}{\sigma^2t+m^2}\bigg)^2V_{xx}-\rho V=0$$

with $V_t$ denoting $V$'s partial derivative with respect to $t$ and $V_{xx}$ denoting $V$'s second partial derivative with respect to $x$.

My closest attempt on finding a solution is the following:

Guess that $V(x,t)=\exp(k\frac{\sigma^2t+m^2}{\sigma^2m}x)$, for some $k\in\mathbb{R}$, so that $V_x=k\frac{\sigma^2t+m^2}{\sigma^2m}V$, $V_{xx}=k^2\bigg(\frac{\sigma^2t+m^2}{\sigma^2m}\bigg)^2V$ and $V_t=k\frac{x}{m}V$, so that plugging back in the PDE, we have

$$k\frac{x}{m}V+\frac{1}{2}\bigg(\frac{\sigma^2 m}{\sigma^2t+m^2}\bigg)^2k^2\bigg(\frac{\sigma^2t+m^2}{\sigma^2m}\bigg)^2V-\rho V=0$$

Or, when $V\neq 0$,

$$k\frac{x}{m}+\frac{1}{2}k^2-\rho =0$$

Which is almost what I would want, in order to determine the constant(s) $k$, but I still have an $x$ multiplying the linear term.

Do you think there's hope for solving the PDE analytically? I feel I'm very close but I'm stuck. Perhaps there's a way of rewriting it through a transformation as the heat equation? If someone could please give me any suggestions on what you think would work to solve it, I would really appreciate it.

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There are separation-of-variables solutions

$$V = a \exp \left(c x + {{\frac {{c}^{2}{m}^{2}{\sigma}^{2}+2\,\rho\,{\sigma}^{2}{t}^ {2}+2\,{m}^{2}\rho\,t}{2\,{\sigma}^{2}t+2\,{m}^{2}}}}\right)$$ where $c$ is an arbitrary complex number. In particular, taking $c$ imaginary leads to solutions

$$ V = a \cos(\omega (x - x_0)) \exp\left( {\frac {1}{{\sigma}^{2}t+{m}^{2}} \left( -{\frac {{\omega}^{2}{m}^{2}{ \sigma}^{2}}{2}}+\rho\,{\sigma}^{2}{t}^{2}+{m}^{2}\rho\,t \right) } \right)$$

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