Next consider 3f∂x + 2f∂y = 0. The dot product of 3,2 with ∇f gives 0, hence f, as a vector function, is constant along any line parallel to 3,2. By the chain rule, g(2x-3y) satisfies the equation for any differentiable function g.
This construction generalizes to higher dimensions and arbitrary coefficients. Consider f∂x+3f∂y+7f∂z = 0. Our function f is constant along every line parallel to 1,3,7. Thus f = g(3x-y,7x-z) for any differentiable function g taking two arguments. Prove this by the chain rule, or add multiples of 1,3,7 to x,y,z and note that the inputs to g don't change, hence f doesn't change, and it satisfies our differential equation.
Of course ordinary and partial differential equations are extensive branches of mathematics; the above examples are merely the tip of the iceberg.