ODE
\[ 2 y(x) y''(x)=6 y'(x)^2+\left (1-3 y(x)^2\right ) y(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.26247 (sec), leaf count = 247
\[\left \{\left \{y(x)\to -\sqrt {-\frac {3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}+3}{-9+4 c_1 \tan ^2(x+c_2)}}\right \},\left \{y(x)\to \sqrt {-\frac {3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}+3}{-9+4 c_1 \tan ^2(x+c_2)}}\right \},\left \{y(x)\to -\sqrt {\frac {-3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}-3}{-9+4 c_1 \tan ^2(x+c_2)}}\right \},\left \{y(x)\to \sqrt {\frac {-3 \tan ^2(x+c_2)+\sqrt {(9+4 c_1) \tan ^2(x+c_2) \sec ^2(x+c_2)}-3}{-9+4 c_1 \tan ^2(x+c_2)}}\right \}\right \}\]
Maple ✓
cpu = 0.386 (sec), leaf count = 37
\[\left [y \left (x \right ) = \frac {1}{\sqrt {-2 \textit {\_C2} \cos \left (x \right )+2 \sin \left (x \right ) \textit {\_C1} +3}}, y \left (x \right ) = -\frac {1}{\sqrt {-2 \textit {\_C2} \cos \left (x \right )+2 \sin \left (x \right ) \textit {\_C1} +3}}\right ]\] Mathematica raw input
DSolve[2*y[x]*y''[x] == y[x]^2*(1 - 3*y[x]^2) + 6*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[-((3 + 3*Tan[x + C[2]]^2 + Sqrt[(9 + 4*C[1])*Sec[x + C[2]]^2*Tan[x + C[2]]^2])/(-9 + 4*C[1]*Tan[x + C[2]]^2))]}, {y[x] -> Sqrt[-((3 + 3*Tan[x + C[2]]^2 + Sqrt[(9 + 4*C[1])*Sec[x + C[2]]^2*Tan[x + C[2]]^2])/(-9 + 4*C[1]*Tan[x + C[2]]^2))]}, {y[x] -> -Sqrt[(-3 - 3*Tan[x + C[2]]^2 + Sqrt[(9 + 4*C[1])*Sec[x + C[2]]^2*Tan[x + C[2]]^2])/(-9 + 4*C[1]*Tan[x + C[2]]^2)]}, {y[x] -> Sqrt[(-3 - 3*Tan[x + C[2]]^2 + Sqrt[(9 + 4*C[1])*Sec[x + C[2]]^2*Tan[x + C[2]]^2])/(-9 + 4*C[1]*Tan[x + C[2]]^2)]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = 6*diff(y(x),x)^2+(1-3*y(x)^2)*y(x)^2, y(x))
Maple raw output
[y(x) = 1/(-2*_C2*cos(x)+2*sin(x)*_C1+3)^(1/2), y(x) = -1/(-2*_C2*cos(x)+2*sin(x)*_C1+3)^(1/2)]