The prompt is to solve for y $$xy' = y + \sqrt{y^2 - x^2}$$
I'm unable to split the equation into parts to integrate them properly, how do you go on about solving such problems?
Here's what I tried doing. $$x\frac{dy}{dx} = y + \sqrt{y^2 + x^2}$$ $$x dy = (y+\sqrt{y^2 + x^2})dx$$ $$xy + C_1 = \frac{1}{2}y\sqrt{y^2(\arcsin{\frac{x}{y}) + \frac{1}{2}\sin(2\arcsin{\frac{x}{y})}}}+yx + C_2$$
But I stopped there, because I wasn't sure if I'm going along the right path.
EDIT(following the comment in answers):
Let $u = \frac{y}{x}$ $$y' = u'x + u$$ $$x(u'x + u) = ux + \sqrt{(ux)^2 - x^2}$$ $$u'x^2 + ux = ux + \sqrt{u^2x^2-x^2}$$ $$\frac{du}{dx}x^2 = x\sqrt{u^2 - 1}$$ $$\int\frac{1}{\sqrt{u^2 - 1}}du = \int\frac{1}{x}dx$$ $$\ln|\sqrt{u^2 - 1} + u| + C_1 = \ln|x| + C_2$$
Hint: Let $u=\dfrac{y}{x}$ then $y=ux$ and $y'=u'x+u$. Substitute and find the solution.