23.4.183 problem 183

Internal problem ID [6485]
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 : 183
Date solved : Tuesday, September 30, 2025 at 03:01:50 PM
CAS classification : [NONE]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=2*y[x]*D[y[x],{x,2}] == -1 + 2*x*f[x]*y[x]^2 - y[x]^4 - 4*y[x]^2*D[y[x],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(-2*x*f(x)*y(x)**2 + y(x)**4 + 4*y(x)**2*Derivative(y(x), x) + 2*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-2*x*f(x)*y(x)**2 + 5*y(x)**4 + 2*y(x)*Derivative(y(x), (x