23.4.20 problem 20

Internal problem ID [6322]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 20
Date solved : Friday, October 03, 2025 at 02:00:44 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 108

ode:=a*y(x)+y(x)*diff(y(x),x)+diff(diff(y(x),x),x) = y(x)^3; 
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. Time used: 64.82 (sec). Leaf size: 3100

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

Too large to display

✗ 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