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I am solving a bunch of differential equations to find the most general form of some functions satisfying the differential equations. I have arrived at my last differential equation which is:

$$\dfrac{\partial f(x,y)}{\partial x} =\dfrac{\partial f(x,y)}{\partial y}$$

Is there any theorem which allows me to give some property about $f$. For instance, I can think of $f(x,y)=f(y,x)$.

Vladimir Vargas
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    It's the advection equation. The solutions are $f(x,y) = h(x+y)$ where $h$ is arbitrary – Tryss Mar 21 '16 at 03:18
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    @Tryss It is also sometimes called the one-way wave equation (since it only propagates in one direction). – Cameron Williams Mar 21 '16 at 03:22
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    Connected with Cameron's comment, the gradient everywhere oriented in the $+45^\circ$ direction. This also implies that all the level sets are of the form $x+y=$const, giving back the property stated by Tryss – Semiclassical Mar 21 '16 at 03:44
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    I appreciate if any of you could post an answer so that the question does not remain unanswered. – Vladimir Vargas Mar 21 '16 at 03:48

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Since nobody answered the question, this is the advection equation, or one-way wave equation, whose solutions are of the form

$$ f(x,y) = h(x+y) $$

for arbitrary $h$.

Vladimir Vargas
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