ODE
\[ \left (y(x)^2+1\right ) y''(x)=3 y(x) y'(x)^2 \] 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 = 0.38113 (sec), leaf count = 93
\[\left \{\left \{y(x)\to -\frac {i c_1 (x+c_2)}{\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}\right \},\left \{y(x)\to \frac {i c_1 (x+c_2)}{\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}\right \}\right \}\]
Maple ✓
cpu = 0.154 (sec), leaf count = 56
\[\left [y \left (x \right ) = \textit {\_C1} x \sqrt {-\frac {1}{\textit {\_C1}^{2} x^{2}+2 \textit {\_C1} x \textit {\_C2} +\textit {\_C2}^{2}-1}}+\sqrt {-\frac {1}{\textit {\_C1}^{2} x^{2}+2 \textit {\_C1} x \textit {\_C2} +\textit {\_C2}^{2}-1}}\, \textit {\_C2}\right ]\] Mathematica raw input
DSolve[(1 + y[x]^2)*y''[x] == 3*y[x]*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> ((-I)*C[1]*(x + C[2]))/Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2]}, {y[x] -> (I*C[1]*(x + C[2]))/Sqrt[-1 + x^2*C[1]^2 + 2*x*C[1]^2*C[2] + C[1]^2*C[2]^2]}}
Maple raw input
dsolve((1+y(x)^2)*diff(diff(y(x),x),x) = 3*y(x)*diff(y(x),x)^2, y(x))
Maple raw output
[y(x) = _C1*x*(-1/(_C1^2*x^2+2*_C1*_C2*x+_C2^2-1))^(1/2)+(-1/(_C1^2*x^2+2*_C1*_C2*x+_C2^2-1))^(1/2)*_C2]