Homogeneous Differential Equation/Examples
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Definition
This page gathers examples of homogeneous differential equations of the first order:
- $\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
where both $M$ and $N$ are homogeneous functions of the same degree.
$\paren {x + y} \d x = \paren {x - y} \d y$
is a homogeneous differential equation with general solution:
- $\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$
$\paren {x^2 - 2 y^2} \d x + x y \rd y = 0$
is a homogeneous differential equation with solution:
- $y^2 = x^2 + C x^4$
$x^2 y' - 3 x y - 2 y^2 = 0$
is a homogeneous differential equation with solution:
- $y = C x^2 \paren {x + y}$
$x^2 y' = 3 \paren {x^2 + y^2} \arctan \dfrac y x + x y$
is a homogeneous differential equation with solution:
- $y = x \tan C x^3$
$x \sin \dfrac y x \dfrac {\d y} {\d x} = y \sin \dfrac y x + x$
is a homogeneous differential equation with solution:
- $\cos \dfrac y x + \ln C x = 0$
$x y' = y + 2 x e^{-y/x}$
is a homogeneous differential equation with solution:
- $e^{y / x} = \ln x^2 + C$