<s>[INST] Solve differential equation $(x^2+xy)y'=x\sqrt{x^2-y^2}+xy+y^2$ Solve and find a particular solution that satisfies $y(1)=1$.
What is the type of this differential equation?
 [/INST] $$(x^2+xy)y'=x\sqrt{x^2-y^2}+xy+y^2$$
$$y'=\dfrac{x\sqrt{x^2-y^2}+xy+y^2}{(x^2+xy)}$$
is an homogeneous differential equation. Let $u=\dfrac{y}{x}$ so
$$u'x+u=\dfrac{\sqrt{1-u^2}+u+u^2}{(u+1)}$$
or
$$\dfrac{u+1}{\sqrt{1-u^2}}du=\dfrac{dx}{x}$$
after integration (let $u=\sin t$) we have
$$t-\cos t=\ln Cx$$
with $y(1)=1$ we get $C=\exp(\dfrac{\pi}{2})$.
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