ODE
\[ \left (y(x)^2+1\right ) y''(x)+(1-2 y(x)) y'(x)^2=0 \] ODE Classification
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.390046 (sec), leaf count = 45
\[\left \{\left \{y(x)\to -\frac {i \left (-1+c_1{}^{2 i} (x+c_2){}^{2 i}\right )}{1+c_1{}^{2 i} (x+c_2){}^{2 i}}\right \}\right \}\]
Maple ✓
cpu = 0.193 (sec), leaf count = 11
\[[y \left (x \right ) = \tan \left (\ln \left (\textit {\_C1} x +\textit {\_C2} \right )\right )]\] Mathematica raw input
DSolve[(1 - 2*y[x])*y'[x]^2 + (1 + y[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*(-1 + C[1]^(2*I)*(x + C[2])^(2*I)))/(1 + C[1]^(2*I)*(x + C[2])^(2*I))}}
Maple raw input
dsolve((1+y(x)^2)*diff(diff(y(x),x),x)+(1-2*y(x))*diff(y(x),x)^2 = 0, y(x))
Maple raw output
[y(x) = tan(ln(_C1*x+_C2))]