23.4.181 problem 181

Internal problem ID [6483]
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 : 181
Date solved : Tuesday, September 30, 2025 at 03:01:49 PM
CAS classification : [[_Painleve, `4th`]]

\begin{align*} 2 y y^{\prime \prime }&=-a^{2}-4 \left (-x^{2}+b \right ) y^{2}+8 x y^{3}+3 y^{4}+{y^{\prime }}^{2} \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a**2 - 8*x*y(x)**3 + (4*b - 4*x**2)*y(x)**2 - 3*y(x)**4 + 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(a**2 + 4*b*y(x)**2 - 4*x**2*y(x)**2 - 8*x*y(x)**3 - 3*y(x)