I read a definition of the differential equation from a book my E.L.Ince as an equation that involves either differentials or differential coefficients. This suggests that a differential equation can have more than one unknown functions(dependent variables) and also more than one independent variables. Can anyone give me an example of a differential equation (not system of equations) that involves more than one unknown dependent variables.
For example, from $\frac{dy}{dx}=y$ and $\frac{d^2 u}{dx^2}=-u$ we can obtain an equation $(\frac{dy}{dx})(\frac{d^2 u}{dx^2})=-yu$. Can we call this a differential equation?
A unit speed curve is one that satisfies the differential equation $$\left({dx\over dt}\right)^2+\left({dy\over dt}\right)^2+\left({dz\over dt}\right)^2=1$$ so there's a differential equation with more than one dependent variable. The heat equation, $${\partial u\over\partial t}-\alpha\left({\partial^2u\over\partial x^2}+{\partial^2u\over\partial y^2}+{\partial^2u\over\partial z^2}\right)=0$$ has more than one independent variable.