23.4.251 problem 251

Internal problem ID [6553]
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 : 251
Date solved : Tuesday, September 30, 2025 at 03:04:22 PM
CAS classification : [NONE]

\begin{align*} 2 \left (1-y\right ) y y^{\prime \prime }&=-\left (1-y\right )^{3} \left (\operatorname {F0} \left (x \right )^{2}-\operatorname {G0} \left (x \right )^{2} y^{2}\right )-4 \left (1-y\right ) y^{2} \left (f \left (x \right )^{2}-g \left (x \right )^{2}+f^{\prime }\left (x \right )+g^{\prime }\left (x \right )\right )-4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=2*(1-y(x))*y(x)*diff(diff(y(x),x),x) = -(1-y(x))^3*(F0(x)^2-G0(x)^2*y(x)^2)-4*(1-y(x))*y(x)^2*(f(x)^2-g(x)^2+diff(f(x),x)+diff(g(x),x))-4*y(x)*(f(x)+g(x)*y(x))*diff(y(x),x)+(1-3*y(x))*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=2*(1 - y[x])*y[x]*D[y[x],{x,2}] == -((1 - y[x])^3*(F0[x]^2 - G0[x]^2*y[x]^2)) - 4*(1 - y[x])*y[x]^2*(f[x]^2 - g[x]^2 + D[f[x],x] + D[g[x],x]) - 4*y[x]*(f[x] + g[x]*y[x])*D[y[x],x] + (1 - 3*y[x])*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(1 - 3*y(x))*Derivative(y(x), x)**2 + (1 - y(x))**3*(F0(x)**2 - G0(x)**2*y(x)**2) + (2 - 2*y(x))*y(x)*Derivative(y(x), (x, 2)) + (4 - 4*y(x))*(f(x)**2 - g(x)**2 + Derivative(f(x), x) + Derivative(g(x), x))*y(x)**2 + 4*(f(x) + g(x)*y(x))*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-2*(f(x) + g(x)*y(x))*y(x) + sqrt(3*F0(x)**2*y(x)**4 - 10*F0(x