<s>[INST] Solving differential equation by separating variables Solve the following differential equation:
$$\frac yx \frac {dy}{dx}=\sqrt {1+x^2+y^2+x^2y^2}$$
I have tried like this:
$$\frac yx \frac{dy}{dx}=\sqrt {1+x^2+y^2(1+x^2)}\\
\implies \frac yx \frac {dy}{dx}=\sqrt {(1+x^2)(1+y^2)}\\
\implies \int \frac {ydy}{\sqrt {1+y^2}}=\int \frac {xdx}{\sqrt{1+x^2}}\\
\implies \log \sqrt{1+y^2}=\log \sqrt {1+x^2}+c_1\\
\implies \frac {1+y^2}{1+x^2}=c\\$$
But in my book the answer is:
$$(1+x^2)^{3/2}-3\sqrt {1+y^2}=c$$
I can't understand how they did it,please check this..
 [/INST] Hint:
Rewrite as below and integrate.
$$\left(\sqrt{1+y^2}\right)^{-1/2}d(y^2)-\left(\sqrt{1+x^2}\right)^{1/2}d(x^2)=0.$$
Mind the asymmetry in the exponents.
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