Definition:Differential Equation/Degree/First
< Definition:Differential Equation | Degree(Redirected from Definition:Quasilinear Differential Equation)
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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