Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Functions with constant divergence of gradient-like field $\phi\nabla \phi/|\nabla \phi|$

I would like to classify functions $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $$ \nabla \cdot \left( \phi \frac{\nabla\phi}{|\nabla\phi|} \right) = \text{const}. $$ The only examples I ...
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Help with a nonlinear partial differential equation

let : $$\frac{\partial f}{\partial x}=f _{x}\;,\;\;\frac{\partial f}{\partial t}=f _{t}\;,\;\;\frac{\partial}{\partial t}\frac{\partial f}{\partial x}=f_{tx}\;, \;\;\ \frac{\partial}{\partial x}\frac{\...