Solution to First Order ODE

Theorem
Let:
 * $\displaystyle \Phi = \frac {\mathrm d y}{\mathrm d x} = f \left({x, y}\right)$

be a first order ordinary differential eqn.

Then $\Phi$ has a general solution which can be expressed in terms of an indefinite integral of $f \left({x}\right)$:
 * $\displaystyle y = \int f \left({x, y}\right) \, \mathrm d x + C$

where $C$ is an arbitrary constant.

Proof
Integrating both sides with respect to $x$:

The validity of this follows from Picard's Existence Theorem.