54.7.31 problem 1621 (6.31)

Internal problem ID [12880]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1621 (6.31)
Date solved : Friday, October 03, 2025 at 03:49:50 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 108

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

\[ -\frac {\int _{}^{y}\frac {4 {\operatorname {RootOf}\left (\left (-4 \textit {\_a}^{6}+12 \textit {\_a}^{4} a -12 \textit {\_a}^{2} a^{2}+4 a^{3}+320 c_1 \right ) \textit {\_Z}^{9}+\left (-189 \textit {\_a}^{6}+567 \textit {\_a}^{4} a -567 \textit {\_a}^{2} a^{2}+189 a^{3}+15120 c_1 \right ) \textit {\_Z}^{6}+238140 c_1 \,\textit {\_Z}^{3}+1250235 c_1 \right )}^{3}+63}{\textit {\_a}^{2}-a}d \textit {\_a}}{63}-x -c_2 = 0 \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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