Solution to First Order ODE

Theorem
Let:
 * $$\Phi = \frac {dy}{dx} = f \left({x, y}\right)$$

be a first order ordinary differential equation.

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

where $$C$$ is an arbitrary constant.

Proof
Integrating both sides with respect to $$x$$:

$$ $$ $$

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