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Suppose that we have the following time-dependent partial differential equation:

\begin{equation} \frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\frac{\partial^2 V(t, x)}{\partial x^2} - wxV(t, x), \quad t> 0, x\in \mathbb{R} \\ V(0, x ) = f(x) = e^{-ux}, \quad x\in \mathbb{R} \end{equation}

How can I come up with $\tilde{V}(t, x)$ as a solution for the above partial differential equation system? Could you please give me the general instruction on how I should proceed to find such a solution from the above partial differential equation?

\begin{equation} \tilde{V}(t, x) = \exp\Big\{\frac{w^2}{6}t^3 + \frac{wu}{2}t^2 +\frac{u^2}{2}t - [wt+u]x\Big\} \end{equation} where $u$ and $w$ are real constants.

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