60.7.178 problem 1798 (book 6.207)

Internal problem ID [11728]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1798 (book 6.207)
Date solved : Sunday, March 30, 2025 at 08:44:52 PM
CAS classification : [[_Painleve, `5th`]]

\begin{align*} 2 x^{2} y \left (-1+y\right ) y^{\prime \prime }-x^{2} \left (3 y-1\right ) {y^{\prime }}^{2}+2 x y \left (-1+y\right ) y^{\prime }+\left (a y^{2}+b \right ) \left (-1+y\right )^{3}+c x y^{2} \left (-1+y\right )+d \,x^{2} y^{2} \left (y+1\right )&=0 \end{align*}

✗ Maple

ode:=2*x^2*y(x)*(-1+y(x))*diff(diff(y(x),x),x)-x^2*(3*y(x)-1)*diff(y(x),x)^2+2*x*y(x)*(-1+y(x))*diff(y(x),x)+(a*y(x)^2+b)*(-1+y(x))^3+c*x*y(x)^2*(-1+y(x))+d*x^2*y(x)^2*(1+y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=c*x*(-1 + y[x])*y[x]^2 + d*x^2*y[x]^2*(1 + y[x]) + (-1 + y[x])^3*(b + a*y[x]^2) + 2*x*(-1 + y[x])*y[x]*D[y[x],x] - x^2*(-1 + 3*y[x])*D[y[x],x]^2 + 2*x^2*(-1 + y[x])*y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
y = Function("y") 
ode = Eq(c*x*(y(x) - 1)*y(x)**2 + d*x**2*(y(x) + 1)*y(x)**2 + 2*x**2*(y(x) - 1)*y(x)*Derivative(y(x), (x, 2)) - x**2*(3*y(x) - 1)*Derivative(y(x), x)**2 + 2*x*(y(x) - 1)*y(x)*Derivative(y(x), x) + (a*y(x)**2 + b)*(y(x) - 1)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - ((y(x) - 1)*y(x) + sqrt(3*a*y(x)**6 - 10*a*y(x)**5 + 12*a*y(x)**4 - 6*a*y(x)**3 + a*y(x)**2 + 3*b*y(x)**4 - 10*b*y(x)**3 + 12*b*y(x)**2 - 6*b*y(x) + b + 3*c*x*y(x)**4 - 4*c*x*y(x)**3 + c*x*y(x)**2 + 3*d*x**2*y(x)**4 + 2*d*x**2*y(x)**3 - d*x**2*y(x)**2 + 6*x**2*y(x)**3*Derivative(y(x), (x, 2)) - 8*x**2*y(x)**2*Derivative(y(x), (x, 2)) + 2*x**2*y(x)*Derivative(y(x), (x, 2)) + y(x)**4 - 2*y(x)**3 + y(x)**2))/(x*(3*y(x) - 1)) cannot be solved by the factorable group method