Quasilinear Differential Equation/Examples/x + y y' = 0/Explicit Solution

Theorem
The first order quasilinear ordinary differential equation:
 * $x + y y' = 0$

has a general solution which can be expressed explicitly as:
 * $y = \pm \sqrt {C - x^2}$

over the domain:
 * $-\sqrt C < x < \sqrt C$

Proof
The general solution of $x + y y' = 0$ is implicitly over the real numbers.

We have:

But when $y = 0$, the derivative $y'$ of the ordinary differential equation $x + y y' = 0$ needs to be infinite for non-zero $x$.

Hence:
 * $-\sqrt C < x < \sqrt C$