23.4.77 problem 77

Internal problem ID [6379]
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 : 77
Date solved : Friday, October 03, 2025 at 02:05:36 AM
CAS classification : [_Emden, [_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{5}+2 y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 75

ode:=x*y(x)^5+2*diff(y(x),x)+x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\operatorname {RootOf}\left (\ln \left (x \right )-6 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {-12 \textit {\_f}^{6}+9 \textit {\_f}^{2}+9 c_1 -3}}d \textit {\_f} +2 c_2 \right )}{\sqrt {x}} \\ y &= \frac {\operatorname {RootOf}\left (\ln \left (x \right )+6 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {-12 \textit {\_f}^{6}+9 \textit {\_f}^{2}+9 c_1 -3}}d \textit {\_f} +2 c_2 \right )}{\sqrt {x}} \\ \end{align*}

✓ Mathematica. Time used: 79.824 (sec). Leaf size: 27478

ode=x*y[x]^5 + 2*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)**5 + x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(-y(x)**5 - Derivative(y(x), (x, 2)))/2 + Derivative(y(x), x)