<s>[INST] Find this ODE solution $(y-x)\sqrt{x^2+1}\dfrac{dy}{dx}=(1+y^2)^{\frac{3}{2}}$ Find all solutions $y$ to the ODE
$$(y-x)\sqrt{x^2+1}\dfrac{dy}{dx}=(1+y^2)^{\frac{3}{2}}$$
My try:
$$(y-x)\dfrac{d(\sqrt{1+y^2})}{y}=\dfrac{(1+y^2)dx}{\sqrt{x^2+1}}$$
then
$$\dfrac{(y-x)\cdot d(\sqrt{1+y^2})}{y(1+y^2)}=\dfrac{d(\sqrt{1+x^2})}{x}$$
then I can't.Thank you for you help 
 [/INST] Maple says:
$$
\arctan \left( x \right) -\int ^{-\arctan \left( x \right) +\arctan
 \left( y \left( x \right)  \right) }\!{\frac {2-\cos \left( 4\,t+4
 \right) -2\,\cos \left( 2\,t \right) +2\,\cos \left( 2\,t+4 \right) -
\cos \left( 4 \right) +\sqrt {4\,\cos \left( 4\,t+8 \right) -2\,\cos
 \left( 8+2\,t \right) -2\,\cos \left( 6\,t+8 \right) -8\,\cos \left( 
4 \right) +16\,\cos \left( 2\,t+4 \right) -12\,\cos \left( 2\,t
 \right) -8\,\cos \left( 4\,t+4 \right) +12}}{2+\cos \left( 4\,t+4
 \right) +2\,\cos \left( 2\,t \right) +2\,\cos \left( 2\,t+4 \right) +
\cos \left( 4 \right) }}{dt}-C=0
$$
</s>