ODE
\[ 3 y(x) y'(x) y''(x)=y'(x)^3-1 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.436115 (sec), leaf count = 126
\[\left \{\left \{y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left (e^{3 c_1} (x+c_2)\right ){}^{3/2}\right )\right \},\left \{y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left (-\sqrt [3]{-1} e^{3 c_1} (x+c_2)\right ){}^{3/2}\right )\right \},\left \{y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left ((-1)^{2/3} e^{3 c_1} (x+c_2)\right ){}^{3/2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.797 (sec), leaf count = 87
\[\left [\frac {3 \left (\textit {\_C1} y \left (x \right )+1\right )^{\frac {2}{3}}}{2 \textit {\_C1}}-x -\textit {\_C2} = 0, \frac {3 \left (\textit {\_C1} y \left (x \right )+1\right )^{\frac {2}{3}}}{\textit {\_C1} \left (-1+i \sqrt {3}\right )}-x -\textit {\_C2} = 0, -\frac {3 \left (\textit {\_C1} y \left (x \right )+1\right )^{\frac {2}{3}}}{\textit {\_C1} \left (1+i \sqrt {3}\right )}-x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[3*y[x]*y'[x]*y''[x] == -1 + y'[x]^3,y[x],x]
Mathematica raw output
{{y[x] -> (-9 + 2*Sqrt[6]*(E^(3*C[1])*(x + C[2]))^(3/2))/(9*E^(3*C[1]))}, {y[x] -> (-9 + 2*Sqrt[6]*(-((-1)^(1/3)*E^(3*C[1])*(x + C[2])))^(3/2))/(9*E^(3*C[1]))}, {y[x] -> (-9 + 2*Sqrt[6]*((-1)^(2/3)*E^(3*C[1])*(x + C[2]))^(3/2))/(9*E^(3*C[1]))}}
Maple raw input
dsolve(3*y(x)*diff(y(x),x)*diff(diff(y(x),x),x) = diff(y(x),x)^3-1, y(x))
Maple raw output
[3/2*(_C1*y(x)+1)^(2/3)/_C1-x-_C2 = 0, 3*(_C1*y(x)+1)^(2/3)/_C1/(-1+I*3^(1/2))-x-_C2 = 0, -3*(_C1*y(x)+1)^(2/3)/_C1/(1+I*3^(1/2))-x-_C2 = 0]