23.4.149 problem 149

Internal problem ID [6451]
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 : 149
Date solved : Tuesday, September 30, 2025 at 02:57:11 PM
CAS classification : [[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

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