<s>[INST] Differential equation $\left(x^2+xy\right)y'=x\sqrt{x^2-y^2}+xy+y^2$ I am not sure which type of differential equation this falls into:
$$\left(x^2+xy\right)y'=x\sqrt{x^2-y^2}+xy+y^2$$ any hints? P.S. I first tried reornazing it so I have $y'$ alone, and hoping that I would get a homogeneous equation, but no such luck.
 [/INST] Divide $x^2$ from both sides and then let $z=\dfrac{y}x$.
Note that $y'=z+xz'$.
The new equation becomes $(1+z)y'=\sqrt{1-z^2}+z+z^2$.
$y'=\sqrt{\dfrac{1-z}{1+z}}+z$
$x\dfrac{\mathrm dz}{\mathrm dx}=\sqrt{\dfrac{1-z}{1+z}}$
$\dfrac{\mathrm dx}{x}=\sqrt{\dfrac{1+z}{1-z}}\ \mathrm dz$
$\ln(x)=\sqrt{\dfrac1{1-z}}\left((z-1)\sqrt{z+1}+2\sqrt{1-z}\sin^{-1}\left(\dfrac{\sqrt{1+z}}{\sqrt2}\right)\right)+C$
And... it gets too complicated.
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