Internal problem ID [6685]
Book: Second order enumerated odes
Section: section 1
Problem number: 52.
ODE order: 2.
ODE degree: 3.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y \left (y^{\prime \prime }\right )^{3}+y^{3} \left (y^{\prime }\right )^{5}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 251
dsolve(y(x)*diff(y(x),x$2)^3+y(x)^3*diff(y(x),x)^5=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{1} \\ \int _{}^{y \relax (x )}\frac {1}{\RootOf \left (-5 \left (\int _{\textit {\_g}}^{\textit {\_Z}}-\frac {1}{\textit {\_a} \left (-\textit {\_f}^{2} \textit {\_a}^{2}\right )^{\frac {1}{3}}-5 \textit {\_f}}d \textit {\_f} \right )-\ln \left (\textit {\_a}^{5}+125\right )+5 c_{1}\right )}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\RootOf \left (-\sqrt {3}\, \ln \left (\textit {\_a}^{5}+125\right )+i \ln \left (\textit {\_a}^{5}+125\right )-20 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {-1+i \sqrt {3}}{\left (5 i \sqrt {3}\, \textit {\_f} -2 \textit {\_a} \left (-\textit {\_f}^{2} \textit {\_a}^{2}\right )^{\frac {1}{3}}-5 \textit {\_f} \right ) \left (\sqrt {3}+i\right )}d \textit {\_f} \right )+20 c_{1}\right )}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\RootOf \left (i \ln \left (\textit {\_a}^{5}+125\right )+\sqrt {3}\, \ln \left (\textit {\_a}^{5}+125\right )-20 \left (\int _{\textit {\_g}}^{\textit {\_Z}}\frac {1+i \sqrt {3}}{\left (5 i \sqrt {3}\, \textit {\_f} +2 \textit {\_a} \left (-\textit {\_f}^{2} \textit {\_a}^{2}\right )^{\frac {1}{3}}+5 \textit {\_f} \right ) \left (i-\sqrt {3}\right )}d \textit {\_f} \right )+20 c_{1}\right )}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.927 (sec). Leaf size: 148
DSolve[y[x]*y''[x]^3+y[x]^3*y'[x]^5==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 0 \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \, _2F_1\left (\frac {3}{5},3;\frac {8}{5};\frac {3 \text {$\#$1}^{5/3}}{5 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \, _2F_1\left (\frac {3}{5},3;\frac {8}{5};-\frac {3 i \left (-i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {27 \text {$\#$1} \, _2F_1\left (\frac {3}{5},3;\frac {8}{5};\frac {3 i \left (i+\sqrt {3}\right ) \text {$\#$1}^{5/3}}{10 c_1}\right )}{c_1{}^3}\&\right ][x+c_2] \\ \end{align*}