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 = 1.10345 (sec), leaf count = 369
\[\left \{\left \{y(x)\to \frac {c_2 \exp \left (-\frac {\tan ^{-1}\left (\frac {1+2 \text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ]\left [c_1-x\right ]{}^2}{\sqrt {3}}\right )}{2 \sqrt {3}}\right )}{\sqrt [4]{\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ]\left [c_1-x\right ]{}^4+\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\& \right ]\left [c_1-x\right ]{}^2+1}}\right \}\right \}\]
Maple ✓
cpu = 0.326 (sec), leaf count = 159
\[ \left \{ \ln \left ( y \left ( x \right ) \right ) -\int \!{\frac {1}{2\,{\it \_C1}+2\,\tan \left ( \sqrt {3}x \right ) } \left ( -\sqrt {3}\tan \left ( \sqrt {3}x \right ) {\it \_C1}+\sqrt {3}-\sqrt { \left ( 3\,{{\it \_C1}}^{2}+4 \right ) \left ( \tan \left ( \sqrt {3}x \right ) \right ) ^{2}+2\,{\it \_C1}\,\tan \left ( \sqrt {3}x \right ) +4\,{{\it \_C1}}^{2}+3} \right ) }\,{\rm d}x-{\it \_C2}=0,\ln \left ( y \left ( x \right ) \right ) -\int \!{\frac {1}{2\,{\it \_C1}+2\,\tan \left ( \sqrt {3}x \right ) } \left ( -\sqrt {3}\tan \left ( \sqrt {3}x \right ) {\it \_C1}+\sqrt {3}+\sqrt { \left ( 3\,{{\it \_C1}}^{2}+4 \right ) \left ( \tan \left ( \sqrt {3}x \right ) \right ) ^{2}+2\,{\it \_C1}\,\tan \left ( \sqrt {3}x \right ) +4\,{{\it \_C1}}^{2}+3} \right ) }\,{\rm d}x-{\it \_C2}=0 \right \} \] 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),'implicit')
Maple raw output
ln(y(x))-Int((-3^(1/2)*tan(3^(1/2)*x)*_C1+3^(1/2)-((3*_C1^2+4)*tan(3^(1/2)*x)^2+2*_C1*tan(3^(1/2)*x)+4*_C1^2+3)^(1/2))/(2*_C1+2*tan(3^(1/2)*x)),x)-_C2 = 0, ln(y(x))-Int((-3^(1/2)*tan(3^(1/2)*x)*_C1+3^(1/2)+((3*_C1^2+4)*tan(3^(1/2)*x)^2+2*_C1*tan(3^(1/2)*x)+4*_C1^2+3)^(1/2))/(2*_C1+2*tan(3^(1/2)*x)),x)-_C2 = 0