Homogeneous differential equation

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A homogeneous differential equation has several distinct meanings.

One meaning is that a first-order ordinary differential equation is homogeneous if it has the form

$\frac{dy}{dx} = F(y/x).$

To solve such equations, one makes the change of variables u = y/x, which will transform such an equation into separable one.

Another meaning is a linear homogeneous differential equation, which is a differential equation of the form

$Ly = 0 \,$

where the differential operator L is a linear operator, and y is the unknown function.

[edit] Example of deriving a homogeneous equation

A well known homogenous equation in x and y of degree m, subsequently showing one of Euler's identities is as follows.

$\ f(x,y) = x^m F\left(\frac{y}{x}\right).$

Deriving $\ f_x (x,y)$ We obtain the following,

$\frac{\partial f(x,y)}{\partial x} = mx^{m-1}F\left(\frac{y}{x}\right)+ x^mF^'\left(\frac{y}{x}\right)\cdot\left(-\frac{y}{x^2}\right)$ .

Where $\ F^'$ denotes the first derivative of F with respect to the homogeneous argument.

Also,

$\frac{\partial f(x,y)}{\partial y} = x^mF^'\left(\frac{y}{x}\right).\left(\frac{1}{x}\right).$

Now taking each derivative and multiplying by its corresponding variable we arrive at the following equation.

$x\frac{\partial f(x,y)}{\partial x} + y\frac{\partial f(x,y)}{\partial y} = x\left[mx^{m-1}F\left(\frac{y}{x}\right)+ x^m.\left(-\frac{y}{x^2}\right)F^'\left(\frac{y}{x}\right)\right] + y\left[ x^m.\left(\frac{1}{x}\right)F^'\left(\frac{y}{x}\right)\right]$

$x\frac{\partial f(x,y)}{\partial x} + y\frac{\partial f(x,y)}{\partial y} = x\left[mx^{m-1}F\left(\frac{y}{x}\right)-x^{m-2}yF^'\left(\frac{y}{x}\right)\right] + y\left[ x^{m-1}F^'\left(\frac{y}{x}\right)\right]$