Question

Show that the differential equation
$x\cos \left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos \left(\frac{y}{x}\right)+x$ is homogeneous.

Answer 1

Lets take the given differential equation,
$x\cos \left(\frac{y}{x}\right)\frac{dy}{dx}=y\cos \left(\frac{y}{x}\right)+x$

Divide by $x\cos \left(\frac{y}{x}\right)$ on both the sides, we get,
$\frac{dy}{dx}=\frac{y\cos \left(\frac{y}{x}\right)+x}{x\cos \left(\frac{y}{x}\right)}$

R.H.S. is a function of $x$ and $y$
$\Rightarrow \frac{dy}{dx}=f(x,y)$

Put $x=\lambda x$ and $y=\lambda y$, we get
$f(\lambda x,\lambda y)={\lambda }^0\left(\frac{y\cos \left(\frac{y}{x}\right)+x}{x\cos \left(\frac{y}{x}\right)}\right)$
$\Rightarrow f(\lambda x,\lambda y)={\lambda }^{0}f(x,y)$

$\Rightarrow f$ is a homogeneous function of degree zero.
Hence, the given differential equation is homogeneous differential equation.