Internal problem ID [9201]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1622 (6.32).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 503
dsolve(diff(diff(y(x),x),x)+(y(x)+3*a)*diff(y(x),x)-y(x)^3+a*y(x)^2+2*a^2*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_f}^{8}-c_{1} \textit {\_f}^{2}+\left (-\textit {\_f}^{12}+2 c_{1} \textit {\_f}^{6}-c_{1}^{2}+\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}\, c_{1} \textit {\_f}^{6}+\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}\, c_{1}^{2}\right )^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1}\right ) \left (-\textit {\_f}^{12}+2 c_{1} \textit {\_f}^{6}-c_{1}^{2}+\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}\, c_{1} \textit {\_f}^{6}+\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}\, c_{1}^{2}\right )^{\frac {1}{3}}}d \textit {\_f} \right ) a +c_{2} a +{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ y \relax (x ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}+\textit {\_f}^{8}+i \sqrt {3}\, c_{1} \textit {\_f}^{2}+i \sqrt {3}\, \left (\left (-\textit {\_f}^{6}+c_{1}\right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )^{\frac {2}{3}}-c_{1} \textit {\_f}^{2}+\left (\left (-\textit {\_f}^{6}+c_{1}\right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1}\right ) \left (\left (-\textit {\_f}^{6}+c_{1}\right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )^{\frac {1}{3}}}d \textit {\_f} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ y \relax (x ) = \RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}-\textit {\_f}^{8}+i \sqrt {3}\, c_{1} \textit {\_f}^{2}+i \sqrt {3}\, \left (\left (-\textit {\_f}^{6}+c_{1}\right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )^{\frac {2}{3}}+c_{1} \textit {\_f}^{2}-\left (\left (-\textit {\_f}^{6}+c_{1}\right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1}\right ) \left (\left (-\textit {\_f}^{6}+c_{1}\right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )^{\frac {1}{3}}}d \textit {\_f} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 22.357 (sec). Leaf size: 88
DSolve[2*a^2*y[x] + a*y[x]^2 - y[x]^3 + (3*a + y[x])*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to {cc} \{ & {cc} \frac {c_1 \wp '(x c_1+c_2;0,1)}{\wp (x c_1+c_2;0,1)} & a=0 \\ -\frac {e^{-a x} c_1 \wp '\left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )}{\wp \left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )} & \text {True} \\ \\ \\ \\ \\ \end{align*}