ODE
\[ \left (y'(x)^2+y(x)^2\right ) y''(x)+y(x)^3=0 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.1377 (sec), leaf count = 369
\[\left \{\left \{y(x)\to \frac {c_2 \exp \left (-\frac {\tan ^{-1}\left (\frac {1+2 \text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ][-x+c_1]{}^2}{\sqrt {3}}\right )}{2 \sqrt {3}}\right )}{\sqrt [4]{\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ][-x+c_1]{}^4+\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ][-x+c_1]{}^2+1}}\right \}\right \}\]
Maple ✓
cpu = 8.518 (sec), leaf count = 297
\[\left [y \left (x \right ) = \left (\textit {\_C1} +\tan \left (\sqrt {3}\, x \right )\right )^{\frac {\textit {\_C1}^{2}}{2 \textit {\_C1}^{2}+2}} \left (1+\tan ^{2}\left (\sqrt {3}\, x \right )\right )^{-\frac {\textit {\_C1}^{2}}{4 \left (\textit {\_C1}^{2}+1\right )}} {\mathrm e}^{\int \frac {\sqrt {3 \textit {\_C1}^{2} \left (\tan ^{2}\left (\sqrt {3}\, x \right )\right )+4 \textit {\_C1}^{2}+2 \textit {\_C1} \tan \left (\sqrt {3}\, x \right )+4 \left (\tan ^{2}\left (\sqrt {3}\, x \right )\right )+3}}{2 \textit {\_C1} +2 \tan \left (\sqrt {3}\, x \right )}d x} \left (\textit {\_C1} +\tan \left (\sqrt {3}\, x \right )\right )^{\frac {1}{2 \textit {\_C1}^{2}+2}} \left (1+\tan ^{2}\left (\sqrt {3}\, x \right )\right )^{-\frac {1}{4 \left (\textit {\_C1}^{2}+1\right )}} \textit {\_C2}, y \left (x \right ) = \left (\textit {\_C1} +\tan \left (\sqrt {3}\, x \right )\right )^{\frac {\textit {\_C1}^{2}}{2 \textit {\_C1}^{2}+2}} \left (1+\tan ^{2}\left (\sqrt {3}\, x \right )\right )^{-\frac {\textit {\_C1}^{2}}{4 \left (\textit {\_C1}^{2}+1\right )}} {\mathrm e}^{\int -\frac {\sqrt {3 \textit {\_C1}^{2} \left (\tan ^{2}\left (\sqrt {3}\, x \right )\right )+4 \textit {\_C1}^{2}+2 \textit {\_C1} \tan \left (\sqrt {3}\, x \right )+4 \left (\tan ^{2}\left (\sqrt {3}\, x \right )\right )+3}}{2 \left (\textit {\_C1} +\tan \left (\sqrt {3}\, x \right )\right )}d x} \left (\textit {\_C1} +\tan \left (\sqrt {3}\, x \right )\right )^{\frac {1}{2 \textit {\_C1}^{2}+2}} \left (1+\tan ^{2}\left (\sqrt {3}\, x \right )\right )^{-\frac {1}{4 \left (\textit {\_C1}^{2}+1\right )}} \textit {\_C2}\right ]\] Mathematica raw input
DSolve[y[x]^3 + (y[x]^2 + y'[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2]/(E^(ArcTan[(1 + 2*InverseFunction[((-I + Sqrt[3])*ArcTan[#1/Sqrt[(1 - I*Sqrt[3])/2]])/Sqrt[6*(1 - I*Sqrt[3])] + ((I + Sqrt[3])*ArcTan[#1/Sqrt[(1 + I*Sqrt[3])/2]])/Sqrt[6*(1 + I*Sqrt[3])] & ][-x + C[1]]^2)/Sqrt[3]]/(2*Sqrt[3]))*(1 + InverseFunction[((-I + Sqrt[3])*ArcTan[#1/Sqrt[(1 - I*Sqrt[3])/2]])/Sqrt[6*(1 - I*Sqrt[3])] + ((I + Sqrt[3])*ArcTan[#1/Sqrt[(1 + I*Sqrt[3])/2]])/Sqrt[6*(1 + I*Sqrt[3])] & ][-x + C[1]]^2 + InverseFunction[((-I + Sqrt[3])*ArcTan[#1/Sqrt[(1 - I*Sqrt[3])/2]])/Sqrt[6*(1 - I*Sqrt[3])] + ((I + Sqrt[3])*ArcTan[#1/Sqrt[(1 + I*Sqrt[3])/2]])/Sqrt[6*(1 + I*Sqrt[3])] & ][-x + C[1]]^4)^(1/4))}}
Maple raw input
dsolve((y(x)^2+diff(y(x),x)^2)*diff(diff(y(x),x),x)+y(x)^3 = 0, y(x))
Maple raw output
[y(x) = (_C1+tan(3^(1/2)*x))^(1/2*_C1^2/(_C1^2+1))*(1+tan(3^(1/2)*x)^2)^(-1/4*_C1^2/(_C1^2+1))*exp(Int(1/(6*_C1+6*tan(3^(1/2)*x))*(3*_C1^2*tan(3^(1/2)*x)^2+4*_C1^2+2*_C1*tan(3^(1/2)*x)+4*tan(3^(1/2)*x)^2+3)^(1/2),x))^3*(_C1+tan(3^(1/2)*x))^(1/2/(_C1^2+1))*(1+tan(3^(1/2)*x)^2)^(-1/4/(_C1^2+1))*_C2, y(x) = (_C1+tan(3^(1/2)*x))^(1/2*_C1^2/(_C1^2+1))*(1+tan(3^(1/2)*x)^2)^(-1/4*_C1^2/(_C1^2+1))/exp(Int(1/(6*_C1+6*tan(3^(1/2)*x))*(3*_C1^2*tan(3^(1/2)*x)^2+4*_C1^2+2*_C1*tan(3^(1/2)*x)+4*tan(3^(1/2)*x)^2+3)^(1/2),x))^3*(_C1+tan(3^(1/2)*x))^(1/2/(_C1^2+1))*(1+tan(3^(1/2)*x)^2)^(-1/4/(_C1^2+1))*_C2]