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 = 0.618433 (sec), leaf count = 78
\[\left \{\left \{y(x)\to -\frac {\sqrt {-2 c_1 x-2 c_2 c_1-1}}{\sqrt {2} \sqrt {c_1 \left (c_2+x\right )}}\right \},\left \{y(x)\to \frac {\sqrt {-2 c_1 x-2 c_2 c_1-1}}{\sqrt {2} \sqrt {c_1 \left (c_2+x\right )}}\right \}\right \}\]
Maple ✓
cpu = 0.13 (sec), leaf count = 23
\[ \left \{ - \left ( 2\, \left ( y \left ( x \right ) \right ) ^{2}+2 \right ) ^{-1}-{\it \_C1}\,x-{\it \_C2}=0 \right \} \] 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),'implicit')
Maple raw output
-1/(2*y(x)^2+2)-_C1*x-_C2 = 0