23.4.267 problem 267

Internal problem ID [6569]
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 : 267
Date solved : Tuesday, September 30, 2025 at 03:07:15 PM
CAS classification : [NONE]

\begin{align*} 2 \left (1-x \right )^{2} x^{2} \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=\operatorname {a0} x \left (1-y\right )^{2} \left (x -y\right )^{2}+\left (-1+\operatorname {a2} \right ) \left (1-x \right ) x \left (1-y\right )^{2} y^{2}+\operatorname {a1} \left (1-x \right ) \left (x -y\right )^{2} y^{2}+\operatorname {a3} \left (1-y\right )^{2} \left (x -y\right )^{2} y^{2}+2 \left (1-x \right ) x \left (1-y\right )^{2} y \left (x^{2}+y-2 x y\right ) y^{\prime }+\left (1-x \right )^{2} x^{2} \left (x -2 y-2 x y+3 y^{2}\right ) {y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=2*(1-x)^2*x^2*(1-y(x))*(x-y(x))*y(x)*diff(diff(y(x),x),x) = a0*x*(1-y(x))^2*(x-y(x))^2+(a2-1)*(1-x)*x*(1-y(x))^2*y(x)^2+a1*(1-x)*(x-y(x))^2*y(x)^2+a3*(1-y(x))^2*(x-y(x))^2*y(x)^2+2*(1-x)*x*(1-y(x))^2*y(x)*(x^2+y(x)-2*x*y(x))*diff(y(x),x)+(1-x)^2*x^2*(x-2*y(x)-2*x*y(x)+3*y(x)^2)*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=2*(1 - x)^2*x^2*(1 - y[x])*(x - y[x])*y[x]*D[y[x],{x,2}] == a0*x*(1 - y[x])^2*(x - y[x])^2 + (-1 + a2)*(1 - x)*x*(1 - y[x])^2*y[x]^2 + a1*(1 - x)*(x - y[x])^2*y[x]^2 + a3*(1 - y[x])^2*(x - y[x])^2*y[x]^2 + 2*(1 - x)*x*(1 - y[x])^2*y[x]*(x^2 + y[x] - 2*x*y[x])*D[y[x],x] + (1 - x)^2*x^2*(x - 2*y[x] - 2*x*y[x] + 3*y[x]^2)*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(-a0*x*(1 - y(x))**2*(x - y(x))**2 - a1*(1 - x)*(x - y(x))**2*y(x)**2 - a3*(1 - y(x))**2*(x - y(x))**2*y(x)**2 + 2*x**2*(1 - x)**2*(1 - y(x))*(x - y(x))*y(x)*Derivative(y(x), (x, 2)) - x**2*(1 - x)**2*(-2*x*y(x) + x + 3*y(x)**2 - 2*y(x))*Derivative(y(x), x)**2 - x*(1 - x)*(1 - y(x))**2*(a2 - 1)*y(x)**2 - x*(1 - y(x))**2*(2 - 2*x)*(x**2 - 2*x*y(x) + 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) - (-x**2*y(x)**3 + 2*x**2*y(x)**2 - x**2*y(x