23.4.235 problem 235

Internal problem ID [6537]
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 : 235
Date solved : Tuesday, September 30, 2025 at 03:02:49 PM
CAS classification : [NONE]

\begin{align*} 4 f \left (x \right ) y y^{\prime \prime }&=4 f \left (x \right )^{2} y+3 f \left (x \right ) g \left (x \right ) y^{2}-f \left (x \right ) y^{4}+2 y^{3} f^{\prime }\left (x \right )+\left (-6 f \left (x \right ) y^{2}+2 f^{\prime }\left (x \right )\right ) y^{\prime }+3 f \left (x \right ) {y^{\prime }}^{2} \end{align*}

✗ Maple

ode:=4*f(x)*y(x)*diff(diff(y(x),x),x) = 4*f(x)^2*y(x)+3*f(x)*g(x)*y(x)^2-f(x)*y(x)^4+2*y(x)^3*diff(f(x),x)+(-6*f(x)*y(x)^2+2*diff(f(x),x))*diff(y(x),x)+3*f(x)*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=4*f[x]*y[x]*D[y[x],{x,2}] == 4*f[x]^2*y[x] + 3*f[x]*g[x]*y[x]^2 - f[x]*y[x]^4 + 2*y[x]^3*D[f[x],x] + (-6*f[x]*y[x]^2 + 2*D[f[x],x])*D[y[x],x] + 3*f[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(-(-6*f(x)*y(x)**2 + 2*Derivative(f(x), x))*Derivative(y(x), x) - 4*f(x)**2*y(x) - 3*f(x)*g(x)*y(x)**2 + f(x)*y(x)**4 + 4*f(x)*y(x)*Derivative(y(x), (x, 2)) - 3*f(x)*Derivative(y(x), x)**2 - 2*y(x)**3*Derivative(f(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(sqrt(-12*f(x)**3*y(x) - 9*f(x)**2*g(x)*y(x)**2 + 12*f(x)**2*y(