Internal problem ID [9196]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1621 (6.31).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }+y^{\prime } y-y^{3}+a y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 108
dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3+a*y(x)=0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}\frac {4 {\operatorname {RootOf}\left (\left (4 \textit {\_a}^{6}-12 a \,\textit {\_a}^{4}+12 \textit {\_a}^{2} a^{2}-4 a^{3}-320 c_{1} \right ) \textit {\_Z}^{9}+\left (189 \textit {\_a}^{6}-567 a \,\textit {\_a}^{4}+567 \textit {\_a}^{2} a^{2}-189 a^{3}-15120 c_{1} \right ) \textit {\_Z}^{6}-238140 c_{1} \textit {\_Z}^{3}-1250235 c_{1} \right )}^{3}+63}{-63 \textit {\_a}^{2}+63 a}d \textit {\_a} -x -c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 35.465 (sec). Leaf size: 990
DSolve[a*y[x] - 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 {$\#$1}}\frac {1}{\frac {e^{6 c_1} \left (a-K[1]^2\right )^2}{2 \sqrt [3]{e^{18 c_1} K[1]^6-3 a e^{18 c_1} K[1]^4+3 a^2 e^{18 c_1} K[1]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[1]^6+3 a e^{30 c_1} K[1]^4-3 a^2 e^{30 c_1} K[1]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[1]^2\right )+\frac {1}{2} e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-3 a e^{18 c_1} K[1]^4+3 a^2 e^{18 c_1} K[1]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[1]^6+3 a e^{30 c_1} K[1]^4-3 a^2 e^{30 c_1} K[1]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} \left (a-K[2]^2\right )^2}{4 \sqrt [3]{e^{18 c_1} K[2]^6-3 a e^{18 c_1} K[2]^4+3 a^2 e^{18 c_1} K[2]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[2]^6+3 a e^{30 c_1} K[2]^4-3 a^2 e^{30 c_1} K[2]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[2]^2\right )-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-3 a e^{18 c_1} K[2]^4+3 a^2 e^{18 c_1} K[2]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[2]^6+3 a e^{30 c_1} K[2]^4-3 a^2 e^{30 c_1} K[2]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} \left (a-K[3]^2\right )^2}{4 \sqrt [3]{e^{18 c_1} K[3]^6-3 a e^{18 c_1} K[3]^4+3 a^2 e^{18 c_1} K[3]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[3]^6+3 a e^{30 c_1} K[3]^4-3 a^2 e^{30 c_1} K[3]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[3]^2\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-3 a e^{18 c_1} K[3]^4+3 a^2 e^{18 c_1} K[3]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[3]^6+3 a e^{30 c_1} K[3]^4-3 a^2 e^{30 c_1} K[3]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[3]\&\right ][x+c_2] \\ \end{align*}