23.4.194 problem 194

Internal problem ID [6496]
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 : 194
Date solved : Tuesday, September 30, 2025 at 03:02:18 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 88

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

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

✓ Mathematica. Time used: 3.485 (sec). Leaf size: 2287

ode=4*y[x]*D[y[x],{x,2}] == a*y[x] + b*y[x]^2 + c*y[x]^3 + 3*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") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-a*y(x) - b*y(x)**2 - c*y(x)**3 + 4*y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE sqrt(3)*sqrt(-(a + b*y(x) + c*y(x)**2 - 4*Derivative(y(x), (x, 2