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?

  • 1
    $\begingroup$ $\sin\frac{x}{x+y}$ is not a differential equation. $\endgroup$ – eloiprime Mar 26 at 4:53
  • $\begingroup$ You should post the Differential equation. $\endgroup$ – Aryadeva Mar 26 at 6:09
  • $\begingroup$ I've editted my post. $\endgroup$ – 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.

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