4.7.4 \(\left (x^2+1\right ) y'(x)=y(x)^2-2 x \left (y(x)^2+1\right ) y(x)+1\)

ODE
\[ \left (x^2+1\right ) y'(x)=y(x)^2-2 x \left (y(x)^2+1\right ) y(x)+1 \] ODE Classification

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica
cpu = 0.542792 (sec), leaf count = 161

\[\text {Solve}\left [c_1=\frac {i \left (x \left (\sqrt [4]{\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {(x+y(x))^2}{(x y(x)-1)^2}\right )-2\right )+y(x) \left (\sqrt [4]{\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {(x+y(x))^2}{(x y(x)-1)^2}\right )+2 x^2\right )\right )}{2 (x y(x)-1) \sqrt [4]{-\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}}},y(x)\right ]\]

Maple
cpu = 0.962 (sec), leaf count = 85

\[ \left \{ {\it \_C1}+{x{\frac {1}{\sqrt [4]{ \left ({x}^{-1}+{{x}^{2} \left ({\frac {{x}^{4}y \relax (x ) }{{x}^{2}+1}}-{\frac {{x}^{3}}{{x}^{2}+1}} \right ) ^{-1}} \right ) ^{2}+1}}}}+{\frac {x+y \relax (x ) }{2\,xy \relax (x ) -2}{\mbox {$_2$F$_1$}({\frac {1}{2}},{\frac {5}{4}};\,{\frac {3}{2}};\,-{\frac { \left (x+y \relax (x ) \right ) ^{2}}{ \left (xy \relax (x ) -1 \right ) ^{2}}})}}=0 \right \} \] Mathematica raw input

DSolve[(1 + x^2)*y'[x] == 1 + y[x]^2 - 2*x*y[x]*(1 + y[x]^2),y[x],x]

Mathematica raw output

Solve[C[1] == ((I/2)*(x*(-2 + Hypergeometric2F1[1/2, 5/4, 3/2, -((x + y[x])^2/(-
1 + x*y[x])^2)]*(((1 + x^2)*(1 + y[x]^2))/(-1 + x*y[x])^2)^(1/4)) + y[x]*(2*x^2 
+ Hypergeometric2F1[1/2, 5/4, 3/2, -((x + y[x])^2/(-1 + x*y[x])^2)]*(((1 + x^2)*
(1 + y[x]^2))/(-1 + x*y[x])^2)^(1/4))))/((-1 + x*y[x])*(-(((1 + x^2)*(1 + y[x]^2
))/(-1 + x*y[x])^2))^(1/4)), y[x]]

Maple raw input

dsolve((x^2+1)*diff(y(x),x) = 1+y(x)^2-2*x*y(x)*(1+y(x)^2), y(x),'implicit')

Maple raw output

_C1+x/((1/x+x^2/(x^4/(x^2+1)*y(x)-x^3/(x^2+1)))^2+1)^(1/4)+(x+y(x))*hypergeom([1
/2, 5/4],[3/2],-(x+y(x))^2/(x*y(x)-1)^2)/(2*x*y(x)-2) = 0