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Let suppose that $a \in \mathbb{R}$ is a real number. For any pair of two real numbers $(x,y) \in \mathbb{R}^2$ we define the function

$G(x,y) = x \cos(xy)\sin(ay) + a \sin(xy)\cos(ay)$

We are looking for a PDE using partial derivatives (possible high order) w.r.t to $x$ and $y$ for which $G(x,y)$ is a solution.

What I've attempted so far: I computed a couple of partial derivatives of $G$ up to the 2nd order but I can't see how to combine them to get a nice PDE.

I don't know if there is a systematic path to follow in order to get that equation.

Pavan C.
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The Wave
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  • You have to be more specific. There are infinitely many PDEs satisfied by $G(x,y)$. For instance, it satisfies the inhomogeneous first order PDE $$\partial_xG(x,y)=\cos(xy)\sin(ay)-xy\sin(xy)\sin(ay)+ay\cos(xy)\cos(ay).$$ – Gonçalo Feb 08 '24 at 05:18
  • Sorry about the clarity issue. We would like it to invole at least a partial derivative w.r.t x and a partial derivative w.r.t y so as to bind both dynamics.

    I don't have any clue if there exists only one PDE or mutiple candidate PDEs. I am actually looking if there is a systmetatic way to identify those quand of PDE starting from a solution.

     – The Wave Feb 08 '24 at 13:25

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