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Consider a PDE of the form \begin{equation} \frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right) \end{equation} or \begin{equation} \frac{\partial^2u}{\partial p\partial t}=\tilde F\left(\frac{\partial u}{\partial p},\frac{\partial u}{\partial t},p\right), \end{equation} ,

where $u$ is twice continuously differentiable. Is there a general way of finding a function $G$ $(\mathrm{or~}\tilde G)$ such that $$\frac{\partial u}{\partial p}=G\left(u\right)$$ or $$\frac{\partial u}{\partial p}=\tilde G\left(\frac{\partial u}{\partial t}\right)~?$$

Background

I have an ODE of the form $$\dot x(t,p) = f(x(t,p),p)$$ and I'm interested in $$\mathrm{sgn}\left(\frac{\partial x}{\partial p}(t,p)\right).$$ We can write the mixed partial derivative in the form of the first equation or, if $f$ is invertible with respect to $x$, in the form of the second equation. Solving the first equation for $\partial_p u$ and assuming non-negativity of $x$ would then be the proposed approach.

Example

Consider the simple decay equation

$$\dot x=-p x.$$

We then have $$\frac{\partial^2x}{\partial p\partial t}=-x-p\frac{\partial x}{\partial p}.$$ From the general solution $$x(t)=x_0 e^{-p t}$$ we know $$\frac{\partial x}{\partial p}=-t x$$ and we can find this using the ansatz $$\frac{\partial x}{\partial p}=h(t,p)x.$$ For much more complicated versions of the function $f$, it is hard to find a working ansatz for the function $h$. Thus, it would be of great help to write $\partial_p x$ in terms of $x$ or $\dot x$ directly from the equation obtained by differentiating $\dot x$ with respect to $p$.

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