Question

$\frac{dy}{dx}=\frac{x^2+y^2}{2xy}$ is a homogeneous Differential equation because

• A. $\frac{(x^2+y^2)}{2xy}$ is not a homogenous function.
• B. It is not a homogenous differential equation.
• C. $\frac{(x^2+y^2)}{2xy}$ is a homogenous function of degree zero.
• D. None of the above.

Answer 1

The correct option is C $\frac{(x^2+y^2)}{2xy}$ is a homogenous function of degree zero.

Remember always that the existence of a homogeneous function is not enough for the differential equation to be homogeneous. That homogeneous function should have degree 0 as well.
Now $\frac{x^2+y^2}{2xy}$. To check its degree replace $x$ by $kx$ and $y$ by $ky$.
So it becomes $\frac{k^2{x}^2+k^2{y}^2}{2k^{2}xy}$
$=\frac{k^2}{2k^2}\frac{(x^2+y^2)}{xy}=\frac{k^0}{2}\frac{(x^2+y^2)}{xy}$
Since $k^0$ is the common factor. So its degree is zero and this is the reason that given equation is a homogeneous differential equation.