<s>[INST] Help solving differential equation: $y' = x\sqrt{4+y^{2}}/{y(9+x^{2})}$ I do not have an idea where to start to solve the following differential equation, so every tip is welcome.
$$y' = \frac{x\sqrt{4+y^{2}}}{y(9+x^{2})} $$
 [/INST] We write your equation as
$$
\frac{y}{\sqrt{4+y^2}}dy=\frac{x}{9+x^2}dx
$$
or equivalenlently,
$$
d(\sqrt{4+y^2})=d((1/2)\ln(9+x^2)).
$$
Hence, we get
$$
\sqrt{4+y^2}=\frac{1}{2}\ln(9+x^2)+C.
$$
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