Internal problem ID [6691]
Book: Second order enumerated odes
Section: section 2
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y^{\prime }+y^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 174
dsolve(diff(y(x),x$2)*diff(y(x),x)+y(x)^n=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{-\frac {\left (\left (-3 \textit {\_a}^{n +1}+c_{1}\right ) \left (n +1\right )^{2}\right )^{\frac {1}{3}}}{2 \left (n +1\right )}-\frac {i \sqrt {3}\, \left (\left (-3 \textit {\_a}^{n +1}+c_{1}\right ) \left (n +1\right )^{2}\right )^{\frac {1}{3}}}{2 \left (n +1\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{-\frac {\left (\left (-3 \textit {\_a}^{n +1}+c_{1}\right ) \left (n +1\right )^{2}\right )^{\frac {1}{3}}}{2 \left (n +1\right )}+\frac {i \sqrt {3}\, \left (\left (-3 \textit {\_a}^{n +1}+c_{1}\right ) \left (n +1\right )^{2}\right )^{\frac {1}{3}}}{2 n +2}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {n +1}{\left (\left (-3 \textit {\_a}^{n +1}+c_{1}\right ) \left (n +1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.501 (sec). Leaf size: 298
DSolve[y''[x]*y'[x]+y[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \, _2F_1\left (\frac {1}{3},\frac {1}{n+1};1+\frac {1}{n+1};\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {(-1)^{2/3} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \, _2F_1\left (\frac {1}{3},\frac {1}{n+1};1+\frac {1}{n+1};\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{-\frac {1}{3}} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \, _2F_1\left (\frac {1}{3},\frac {1}{n+1};1+\frac {1}{n+1};\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-\text {$\#$1}^{n+1}+c_1 (n+1)}}\&\right ][x+c_2] \\ \end{align*}