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 = 0.246352 (sec), leaf count = 153
\[\left \{\left \{y(x)\to \frac {1}{9} e^{-3 c_1} \left (2 \sqrt {6} \left (e^{3 c_1} \left (c_2+x\right )\right ){}^{3/2}-9\right )\right \},\left \{y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9-(-1)^{2/3} \sqrt {2} \left (\sqrt {3}+3 i\right ) \left (-\sqrt [3]{-1} e^{3 c_1} \left (c_2+x\right )\right ){}^{3/2}\right )\right \},\left \{y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+\sqrt [3]{-1} \sqrt {2} \left (\sqrt {3}-3 i\right ) \left ((-1)^{2/3} e^{3 c_1} \left (c_2+x\right )\right ){}^{3/2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.085 (sec), leaf count = 87
\[ \left \{ {\frac {3}{2\,{\it \_C1}} \left ( {\it \_C1}\,y \left ( x \right ) +1 \right ) ^{{\frac {2}{3}}}}-x-{\it \_C2}=0,3\,{\frac { \left ( {\it \_C1}\,y \left ( x \right ) +1 \right ) ^{2/3}}{{\it \_C1}\, \left ( i\sqrt {3}-1 \right ) }}-x-{\it \_C2}=0,-3\,{\frac { \left ( {\it \_C1}\,y \left ( x \right ) +1 \right ) ^{2/3}}{{\it \_C1}\, \left ( i\sqrt {3}+1 \right ) }}-x-{\it \_C2}=0 \right \} \] 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 - (-1)^(2/3)*Sqrt[2]*(3*I + Sqrt[3])*(-((-1)^(1/3)*E^(3*C[1])*(x + C[2])))^(3/2))/(9*E^(3*C[1]))}, {y[x] -> (-9 + (-1)^(1/3)*Sqrt[2]*(-3*I + Sqrt[3])*((-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),'implicit')
Maple raw output
3/2*(_C1*y(x)+1)^(2/3)/_C1-x-_C2 = 0, 3*(_C1*y(x)+1)^(2/3)/_C1/(I*3^(1/2)-1)-x-_C2 = 0, -3*(_C1*y(x)+1)^(2/3)/_C1/(I*3^(1/2)+1)-x-_C2 = 0