23.4.258 problem 258

Internal problem ID [6560]
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 : 258
Date solved : Tuesday, September 30, 2025 at 03:04:26 PM
CAS classification : [NONE]

\begin{align*} \operatorname {a2} x \left (1-y\right ) y^{2}+\operatorname {a3} \,x^{3} y^{2} \left (1+y\right )+\left (1-y\right )^{3} \left (\operatorname {a0} +\operatorname {a1} y^{2}\right )+2 x \left (1-y\right ) y y^{\prime }-x^{2} \left (1-3 y\right ) {y^{\prime }}^{2}+2 x^{2} \left (1-y\right ) y y^{\prime \prime }&=0 \end{align*}

✗ Maple

ode:=a2*x*(1-y(x))*y(x)^2+a3*x^3*y(x)^2*(1+y(x))+(1-y(x))^3*(a0+a1*y(x)^2)+2*x*(1-y(x))*y(x)*diff(y(x),x)-x^2*(1-3*y(x))*diff(y(x),x)^2+2*x^2*(1-y(x))*y(x)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=a2*x*(1 - y[x])*y[x]^2 + a3*x^3*y[x]^2*(1 + y[x]) + (1 - y[x])^3*(a0 + a1*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") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(a2*x*(1 - y(x))*y(x)**2 + a3*x**3*(y(x) + 1)*y(x)**2 - x**2*(1 - 3*y(x))*Derivative(y(x), x)**2 + 2*x**2*(1 - y(x))*y(x)*Derivative(y(x), (x, 2)) + 2*x*(1 - y(x))*y(x)*Derivative(y(x), x) + (1 - y(x))**3*(a0 + a1*y(x)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - ((y(x) - 1)*y(x) + sqrt(3*a0*y(x)**4 - 10*