ODE
\[ y(x) \left (y(x)^2+1\right ) y''(x)+\left (1-3 y(x)^2\right ) y'(x)^2=0 \] 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 = 0.703652 (sec), leaf count = 78
\[\left \{\left \{y(x)\to -\frac {\sqrt {-2 c_1 x-1-2 c_2 c_1}}{\sqrt {2} \sqrt {c_1 (x+c_2)}}\right \},\left \{y(x)\to \frac {\sqrt {-2 c_1 x-1-2 c_2 c_1}}{\sqrt {2} \sqrt {c_1 (x+c_2)}}\right \}\right \}\]
Maple ✓
cpu = 0.138 (sec), leaf count = 61
\[\left [y \left (x \right ) = -\frac {\sqrt {-2 \left (\textit {\_C1} x +\textit {\_C2} \right ) \left (2 \textit {\_C1} x +2 \textit {\_C2} +1\right )}}{2 \left (\textit {\_C1} x +\textit {\_C2} \right )}, y \left (x \right ) = \frac {\sqrt {-2 \left (\textit {\_C1} x +\textit {\_C2} \right ) \left (2 \textit {\_C1} x +2 \textit {\_C2} +1\right )}}{2 \textit {\_C1} x +2 \textit {\_C2}}\right ]\] Mathematica raw input
DSolve[(1 - 3*y[x]^2)*y'[x]^2 + y[x]*(1 + y[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[-1 - 2*x*C[1] - 2*C[1]*C[2]]/(Sqrt[2]*Sqrt[C[1]*(x + C[2])]))}, {y[x] -> Sqrt[-1 - 2*x*C[1] - 2*C[1]*C[2]]/(Sqrt[2]*Sqrt[C[1]*(x + C[2])])}}
Maple raw input
dsolve(y(x)*(1+y(x)^2)*diff(diff(y(x),x),x)+(1-3*y(x)^2)*diff(y(x),x)^2 = 0, y(x))
Maple raw output
[y(x) = -1/2*(-2*(_C1*x+_C2)*(2*_C1*x+2*_C2+1))^(1/2)/(_C1*x+_C2), y(x) = 1/2*(-2*(_C1*x+_C2)*(2*_C1*x+2*_C2+1))^(1/2)/(_C1*x+_C2)]