Solve the differential equation $\frac {du}{dx} \times \frac {du}{dy}= 0$

Question

I saw this problem and I am not sure how should I start to solve it. Could you please help me with some hints?

Solve the following equation for functions u ∈ $C^2(R^2, R)$

$d_{1}u \times d_{2}u=0$

I know that one or both of them should be zero, meaning it is a constant but I am clueless about how can I use it.

Answer 1

Some more information about what the question is asking would be useful.

Are you solving for $u$?

What is $u$ a function of?

If $u$ is a function of $x$ and $y$, then perhaps the equation should read $\frac{\partial u}{\partial x}\cdot\frac{\partial u}{\partial y}=0$?