First-order first-degree differential equation

Definition

In unnormalized form

A first-order first-degree differential equation is a differential equation that is both a first-order differential equation and a first-degree differential equation. Explicitly, it has the form:

$P(x,y){\frac {dy}{dx}}=Q(x,y)$

where $P,Q$ are known functions. Here, $x$ is the independent variable and $y$ is the dependent variable.

In normalized form

In normalized form, a first-order first-degree differential equation can be written as:

${\frac {dy}{dx}}=G(x,y)$

Conversion between the forms

A first-order first-degree differential equation can be converted to normalized form as follows. Start with:

$P(x,y){\frac {dy}{dx}}=Q(x,y)$

Now, divide both sides by $P(x,y)$ and set $G(x,y):=Q(x,y)/P(x,y)$, giving the normalized form.

Note that the process may involve some slight change in the set of solutions. In particular, any solution that identically satisfies both $P(x,y)=0$ and $Q(x,y)=0$ may be lost when we normalize. In most cases, there are no such solutions, and there are usually at most finitely many such solutions.

Existence and uniqueness of solutions

• Peano existence theorem guarantees that existence of a local solution to any initial value problem for a first-order first-degree differential equations ${\frac {dy}{dx}}=G(x,y)$ with initial value point $(x_{0},y_{0})$ provided that $G$ and $\partial G/\partial y$ are continuous in an open rectangle containing the point.
• Picard-Lindelof theorem establishes existence and uniqueness under somewhat stronger continuity and differentiability assumptions.