23.4.175 problem 175

Internal problem ID [6477]
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 : 175
Date solved : Tuesday, September 30, 2025 at 03:01:45 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y y^{\prime \prime }&=4 y^{2}+8 y^{3}+{y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 63

ode:=2*y(x)*diff(diff(y(x),x),x) = 4*y(x)^2+8*y(x)^3+diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ \int _{}^{y}\frac {1}{\sqrt {\left (4 \textit {\_a}^{2}+c_1 +4 \textit {\_a} \right ) \textit {\_a}}}d \textit {\_a} -x -c_2 &= 0 \\ -\int _{}^{y}\frac {1}{\sqrt {\left (4 \textit {\_a}^{2}+c_1 +4 \textit {\_a} \right ) \textit {\_a}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.498 (sec). Leaf size: 1095

ode=2*y[x]*D[y[x],{x,2}] == 4*y[x]^2 + 8*y[x]^3 + D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x)**3 - 4*y(x)**2 + 2*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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