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 $x$:

\(\ds \int \paren {\frac {\d y} {\d x} } \rd x\)

\(=\)

\(\ds \int \map f {x, y} \rd x\)

\(\ds \leadsto \ \ \)

\(\ds y + C_1\)

\(=\)

\(\ds \int \map f {x, y} \rd x\)

Definition of Indefinite Integral: $C_1$ is arbitrary

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds \int \map f {x, y} \rd x + C\)

replacing $-C_1$ with $C$

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

$\blacksquare$