ODE
\[ y(x) y'''(x)+y(x)^3 y'(x)-y'(x) y''(x)=0 \] ODE Classification
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.40019 (sec), leaf count = 409
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right )|\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\& \right ][x+c_3]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right )|\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\& \right ][x+c_3]\right \}\right \}\]
Maple ✓
cpu = 2.418 (sec), leaf count = 77
\[\left [\int _{}^{y \left (x \right )}-\frac {2}{\sqrt {-\textit {\_a}^{4}+4 \textit {\_C2} \,\textit {\_a}^{2}-4 \textit {\_C2}^{2}+4 \textit {\_C1}}}d \textit {\_a} -x -\textit {\_C3} = 0, \int _{}^{y \left (x \right )}\frac {2}{\sqrt {-\textit {\_a}^{4}+4 \textit {\_C2} \,\textit {\_a}^{2}-4 \textit {\_C2}^{2}+4 \textit {\_C1}}}d \textit {\_a} -x -\textit {\_C3} = 0\right ]\] Mathematica raw input
DSolve[y[x]^3*y'[x] - y'[x]*y''[x] + y[x]*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[((-2*I)*EllipticF[I*ArcSinh[(Sqrt[(-C[2] + Sqrt[-C[1] + C[2]^2])^(-1)]*#1)/Sqrt[2]], (C[2] - Sqrt[-C[1] + C[2]^2])/(C[2] + Sqrt[-C[1] + C[2]^2])]*Sqrt[1 + #1^2/(2*(-C[2] + Sqrt[-C[1] + C[2]^2]))]*Sqrt[1 - #1^2/(2*(C[2] + Sqrt[-C[1] + C[2]^2]))])/(Sqrt[(-C[2] + Sqrt[-C[1] + C[2]^2])^(-1)]*Sqrt[-2*C[1] + 2*C[2]*#1^2 - #1^4/2]) & ][x + C[3]]}, {y[x] -> InverseFunction[((2*I)*EllipticF[I*ArcSinh[(Sqrt[(-C[2] + Sqrt[-C[1] + C[2]^2])^(-1)]*#1)/Sqrt[2]], (C[2] - Sqrt[-C[1] + C[2]^2])/(C[2] + Sqrt[-C[1] + C[2]^2])]*Sqrt[1 + #1^2/(2*(-C[2] + Sqrt[-C[1] + C[2]^2]))]*Sqrt[1 - #1^2/(2*(C[2] + Sqrt[-C[1] + C[2]^2]))])/(Sqrt[(-C[2] + Sqrt[-C[1] + C[2]^2])^(-1)]*Sqrt[-2*C[1] + 2*C[2]*#1^2 - #1^4/2]) & ][x + C[3]]}}
Maple raw input
dsolve(y(x)*diff(diff(diff(y(x),x),x),x)-diff(y(x),x)*diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x) = 0, y(x))
Maple raw output
[Intat(-2/(-_a^4+4*_C2*_a^2-4*_C2^2+4*_C1)^(1/2),_a = y(x))-x-_C3 = 0, Intat(2/(-_a^4+4*_C2*_a^2-4*_C2^2+4*_C1)^(1/2),_a = y(x))-x-_C3 = 0]