Solution to First Order ODE

Theorem
Let:
 * $\Phi = \dfrac {\d y} {\d x} = \map f {x, y}$

be a first order ordinary differential equation.

Then $\Phi$ has a general solution which can be expressed in terms of an indefinite integral of $\map f x$:
 * $\ds y = \int \map f {x, y} \rd x + C$

where $C$ is an arbitrary constant.

Proof
Integrating both sides with respect to $x$:

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