23.4.188 problem 188

Internal problem ID [6490]
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 : 188
Date solved : Tuesday, September 30, 2025 at 03:01:56 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y y^{\prime \prime }&=-y^{2} \left (1+a y^{3}\right )+6 {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 77

ode:=2*y(x)*diff(diff(y(x),x),x) = -y(x)^2*(1+a*y(x)^3)+6*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ -2 \int _{}^{y}\frac {1}{\sqrt {4 c_1 \,\textit {\_a}^{4}+4 \textit {\_a}^{3} a +1}\, \textit {\_a}}d \textit {\_a} -x -c_2 &= 0 \\ 2 \int _{}^{y}\frac {1}{\sqrt {4 c_1 \,\textit {\_a}^{4}+4 \textit {\_a}^{3} a +1}\, \textit {\_a}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 29.561 (sec). Leaf size: 2761

ode=2*y[x]*D[y[x],{x,2}] == -(y[x]^2*(1 + a*y[x]^3)) + 6*D[y[x],x]^2; 
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)**3 + 1)*y(x)**2 + 2*y(x)*Derivative(y(x), (x, 2)) - 6*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(6)*sqrt((a*y(x)**4 + y(x) + 2*Derivative(y(x), (x, 2)))*y(