Homogeneous Functions | Equations of Order One

If the function f(x, y) remains unchanged after replacing x by kx and y by ky, where k is a constant term, then f(x, y) is called a homogeneous function. A differential equation
 

$M \, dx + N \, dy = 0 \,\, \to \,\,$ Equation (1)

 
is homogeneous in x and y if M and N are homogeneous functions of the same degree in x and y.
 

To solve for Equation (1) let
 

$y = vx \,\, \text{and} \,\, dy = v \, dx + x\, dv$

or

$x = vy \,\, \text{and} \,\, dx = v \, dy + y\, dv$

 
The substitution above will lead to variables separable differential equation.
 

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