# Which of the following is a homogeneous differential equation?

$\begin{array}{1 1} (A)\quad(4x+6y+5)\;dy-(3y+2x+4)\;dx = 0\\(B)\quad(xy)\;dx-(x^3+y^3)\;dy = 0\\(C)\quad(x^3+2y^2)dx+2xy\;dy=0\\(D)\quad y^2dx+(x^2-xy-y^2)\;dy=0\end{array}$

Toolbox:
• A differential equation of the form $dy/dx = F(x,y)$ is said to be homogenous, if $F(x,y)$ is a homogenous function of degree zero.
Using the information in the tool box, if the equations are expressed in the form of $F(kx,ky)$, we get only for option D as a function of zero.
Writing the equation of option D as $dy/dx = y^2/(x^2 -xy-y^2)$
i.e;$F(x,y) = y^2/(x^2 -xy -y^2)$
$F(kx,ky) =k^2x^2/(k^2x^2 - kxky - k^2y^2) = k^0.F(x,y)$
Hence it is a homogenous function with degree zero.
edited Jul 8, 2013