23.4.202 problem 202

Internal problem ID [6504]
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 : 202
Date solved : Tuesday, September 30, 2025 at 03:02:26 PM
CAS classification : [[_Painleve, `3rd`]]

\begin{align*} x y y^{\prime \prime }&=y \left (\operatorname {a2} +\operatorname {a3} y^{2}\right )+x \left (\operatorname {a0} +\operatorname {a1} y^{4}\right )-y y^{\prime }+x {y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=x*y(x)*diff(diff(y(x),x),x) = y(x)*(a2+a3*y(x)^2)+x*(a0+a1*y(x)^4)-y(x)*diff(y(x),x)+x*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=x*y[x]*D[y[x],{x,2}] == y[x]*(a2 + a3*y[x]^2) + x*(a0 + a1*y[x]^4) - y[x]*D[y[x],x] + x*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(-x*(a0 + a1*y(x)**4) + x*y(x)*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x)**2 - (a2 + a3*y(x)**2)*y(x) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-4*a0*x**2 - 4*a1*x**2*y(x)**4 - 4*a