Question

The degree of the homogeneous function $x^2\sin \left(\frac{y^2}{x^2}\right)+y^2$is ___

Answer 1

So how to check the degree of a homogeneous function? We discussed that $f(x,y)$ has to be written as $f(kx,ky)$ first. So let’s do that.
$f(x,y)=x^2\left(\sin \left(\frac{y^2}{x^2}\right)\right)+y^{2}f(kx,ky)=k^2{x}^2\left(\sin \left(\frac{k^2{y}^2}{k^2{x}^2}\right)\right)+k^2{y}^2=k^2(x^2\sin \left(\frac{y^2}{x^2}\right)+y^2)$
So we can notice that after multiplying each $x$ and $y$ with $k$ we can rewrite the function in such a manner that we can take $k^2$ out as common leaving every $x$ and $y$ independent of $k$.
Here $k^2$ is common. Rest of the function is independent of $k$. Now degree of a homogeneous function is that power of $k$ which was taken out as common factor. Here its $k^2$. So degree of the given homogeneous function is 2.