Suppose I have three suitably-smooth real-valued function $x(t), y(t), z(t)$ where $t \in \mathbb{R}$. Let's consider the following differential equation:
$$\left( \frac{\partial x(t)}{\partial t} \right)^2 + \left( \frac{\partial y(t)}{\partial t} \right)^2 + \left( \frac{\partial z(t)}{\partial t} \right)^2 = C$$
where $C$ is real-valued constant.
Clearly this isn't the heat equation, wave equation, or Laplace's equation. The first difficulty prima facie is the non-linearity. Taking $u := \frac{\partial x}{\partial t}$, $v := \frac{\partial y}{\partial t}$, and $w := \frac{\partial z}{\partial t}$ we have the algebraic expression:
$$u^2 + v^2 + w^2 = C.$$
Taking the first derivative wrt $t$, we obtain:
$$2u \frac{\partial u}{\partial t} + 2v\frac{\partial v}{\partial t} + 2w\frac{\partial w}{\partial t} = 0$$
via the chain rule. Dividing both sides by 2, we obtain
$$u \frac{\partial u}{\partial t} + v\frac{\partial v}{\partial t} + w\frac{\partial w}{\partial t} = 0$$
But now looking at this I am not sure how to proceed. I considered Laplace transforming the expression, but that seems to lead to an integral I don't know how to solve.
How can I proceed?