Solution to First Order ODE
Jump to navigation
Jump to search
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.
$\blacksquare$