# Is $\sin\frac{x}{x+y}$ homogeneous or not? [closed]

First time I looked up this problem, I say yes but now I'm confused.

Consider the following 1st order differential equation, $$\frac{dy}{dx}=\sin\frac{x}{x+y}.$$

Substitute $$v=y/x$$ then it leads to a 1st order separable equation. Is this wrong?

• $\sin\frac{x}{x+y}$ is not a differential equation. – eloiprime Mar 26 at 4:53
• You should post the Differential equation. – Aryadeva Mar 26 at 6:09
• I've editted my post. – Sh7 Mar 26 at 6:16

A function $$f(x,y)$$ is homogeneous if $$f(tx,ty)=t^k f(x,y)$$, $$k$$ is any real number and is the degree of the homogeneous equation.
Here, $$f(x,y)=\sin \left(\dfrac{x}{x+y} \right)$$. Now $$f(tx,ty)=\sin \left(\dfrac{tx}{t(x+y)} \right)=\sin \left(\dfrac{x}{x+y} \right)$$.
Hence the function $$\sin \left(\dfrac{x}{x+y} \right)$$ is homogeneous.