Internal problem ID [9943]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1620 (6.30).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }+y^{\prime } y-y^{3}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 336
dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3=0,y(x), singsol=all)
\begin{align*} 2 \left (\int _{}^{y \left (x \right )}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \textit {\_a}^{4}+i \sqrt {3}\, \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 4 \left (\int _{}^{y \left (x \right )}-\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \textit {\_a}^{4}-i \sqrt {3}\, \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 46.156 (sec). Leaf size: 1534
DSolve[-y[x]^3 + y[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2}{\frac {e^{6 c_1} K[1]^4}{\sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}-K[1]^2+e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} K[2]^4}{4 \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} K[3]^4}{4 \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2}{\frac {e^{6 (-c_1)} K[1]^4}{\sqrt [3]{e^{18 (-c_1)} K[1]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[1]^6}}}-K[1]^2+e^{-6 (-c_1)} \sqrt [3]{e^{18 (-c_1)} K[1]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[1]^6}}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2}{\frac {e^{6 c_1} K[1]^4}{\sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}-K[1]^2+e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 (-c_1)} K[2]^4}{4 \sqrt [3]{e^{18 (-c_1)} K[2]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 (-c_1)} \sqrt [3]{e^{18 (-c_1)} K[2]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[2]^6}}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} K[2]^4}{4 \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 (-c_1)} K[3]^4}{4 \sqrt [3]{e^{18 (-c_1)} K[3]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 (-c_1)} \sqrt [3]{e^{18 (-c_1)} K[3]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[3]^6}}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} K[3]^4}{4 \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}dK[3]\&\right ][x+c_2] \\ \end{align*}