Internal problem ID [7447]
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]]
\[ \boxed {y^{\prime \prime } y^{\prime }+y^{n}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 174
dsolve(diff(y(x),x$2)*diff(y(x),x)+y(x)^n=0,y(x), singsol=all)
\begin{align*} \frac {\left (-2-2 n \right ) \left (\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right )-\left (1+i \sqrt {3}\right ) \left (x +c_{2} \right )}{1+i \sqrt {3}} &= 0 \\ -\frac {2 i \left (1+n \right ) \left (\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right )+\left (x +c_{2} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \left (\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right ) n +\int _{}^{y \left (x \right )}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_{1} \right ) \left (1+n \right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} -c_{2} -x &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.4 (sec). Leaf size: 910
DSolve[y''[x]*y'[x]+y[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {(-1)^{2/3} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{-\frac {1}{3}} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-\text {$\#$1}^{n+1}+c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{(-c_1) (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) (-c_1)}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 (-c_1) (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {(-1)^{2/3} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{(-c_1) (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) (-c_1)}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 (-c_1) (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{-\frac {1}{3}} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{(-c_1) (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) (-c_1)}\right )}{\sqrt [3]{-\text {$\#$1}^{n+1}+(-c_1) (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {(-1)^{2/3} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-3 \text {$\#$1}^{n+1}+3 c_1 (n+1)}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{-\frac {1}{3}} \text {$\#$1} \sqrt [3]{n+1} \sqrt [3]{1-\frac {\text {$\#$1}^{n+1}}{c_1 (n+1)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{n+1},1+\frac {1}{n+1},\frac {\text {$\#$1}^{n+1}}{(n+1) c_1}\right )}{\sqrt [3]{-\text {$\#$1}^{n+1}+c_1 (n+1)}}\&\right ][x+c_2] \\ \end{align*}