Definition:Differential Equation/Degree/First

Definition

A quasilinear differential equation is a differential equation of the first degree.

Examples

First Order Quasilinear Ordinary Differential Equation

A first order quasilinear ordinary differential equation is a differential equation which can be written in the form:

$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$

First Order Quasilinear ODE: $x + y y' = 0$

The first order quasilinear ordinary differential equation over the real numbers $\R$:

$x + y y' = 0$

has the general solution:

$x^2 + y^2 = C$

where:

$C > 0$

$y \ne 0$

$x < \size {\sqrt C}$

with the singular point:

$x = y = 0$

Also see

• Results about quasilinear differential equations can be found here.

Sources

• 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction