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Let be a function $u\in\mathcal{C}^{\infty}(\mathbb{R}^2)$. Then, from a geometric problem I've obtained that $u$ satisfy the following PDE's (in all its domain):

$\Delta u=0$ (harmonic functions),

$\dfrac{\partial u}{\partial x\partial y}=\dfrac{\partial u}{\partial x}\cdot\dfrac{\partial u}{\partial y}.$

I want to know what functions are a solution to these PDE's. I don't know how to solve them but I do know a family of solutions, the family given by

$u(x,y)=ax+b$ with $a,b\in\mathbb{R}$ and and the same with the variable $y$.

Any comments would be appreciated.

Ruyman
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  • The first equations show you that you can use variable separation, u(x,y)=A(x)+B(y), this will cause your second differential equation to be annuled and you'll have two possible situations, with which you'll be able to solve different ordinary differential equations for A and B – Cristian Rodríguez Feb 28 '21 at 23:33
  • Sorry, I don't know if it's trivial and I'm not seeing it or it's not trivial. Why can I do separation of variables? Of course the Laplacian equation alone does not imply it. – Ruyman Mar 01 '21 at 15:12
  • I don't remember why, but in order to use the method of separation of variables we must be working with a linear homogenous partial differential equations with linear homogeneous boundary conditions. – Cristian Rodríguez Mar 02 '21 at 19:43

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