Internal problem ID [9212]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1633.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime } y a +b y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 97
dsolve(diff(diff(y(x),x),x)+a*y(x)*diff(y(x),x)+b*y(x)^3=0,y(x), singsol=all)
\[ \int _{}^{y \relax (x )}\frac {1}{\RootOf \left (-2 a \,\textit {\_a}^{2} \arctanh \left (\frac {\textit {\_a}^{2} a +4 \textit {\_Z}}{\sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}}\right )-\ln \left (\textit {\_a}^{4} b +\textit {\_Z} \,\textit {\_a}^{2} a +2 \textit {\_Z}^{2}\right ) \sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}+c_{1} \sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}\right )}d \textit {\_a} -x -c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 28.505 (sec). Leaf size: 92
DSolve[b*y[x]^3 + a*y[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{K[2]^2 \text {InverseFunction}\left [\frac {1}{4} \left (\log (b+\text {$\#$1} (a+2 \text {$\#$1}))-\frac {2 a \text {ArcTan}\left (\frac {a+4 \text {$\#$1}}{\sqrt {8 b-a^2}}\right )}{\sqrt {8 b-a^2}}\right )\&\right ][c_1-\log (K[2])]}dK[2]=x-c_2,y(x)\right ] \]