54.7.38 problem 1633 (6.43)

Internal problem ID [12887]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1633 (6.43)
Date solved : Wednesday, October 01, 2025 at 02:44:43 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+a y y^{\prime }+b y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 97

ode:=diff(diff(y(x),x),x)+a*y(x)*diff(y(x),x)+b*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (-2 a \,\textit {\_a}^{2} \operatorname {arctanh}\left (\frac {\textit {\_a}^{2} a +4 \textit {\_Z}}{\sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}}\right )-\ln \left (\textit {\_a}^{4} b +\textit {\_Z} \,\textit {\_a}^{2} a +2 \textit {\_Z}^{2}\right ) \sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}+c_1 \sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}\right )}d \textit {\_a} -x -c_2 = 0 \]

✓ Mathematica. Time used: 25.157 (sec). Leaf size: 92

ode=b*y[x]^3 + a*y[x]*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{K[2]^2 \text {InverseFunction}\left [\frac {1}{4} \left (\log (b+\text {$\#$1} (a+2 \text {$\#$1}))-\frac {2 a \arctan \left (\frac {a+4 \text {$\#$1}}{\sqrt {8 b-a^2}}\right )}{\sqrt {8 b-a^2}}\right )\&\right ][c_1-\log (K[2])]}dK[2]=x-c_2,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), x) + b*y(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x)**3 - Derivative(y(x), (x, 2)))/(a*y(x)) cannot be solved by the factorable group method