23.4.159 problem 159

Internal problem ID [6461]
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 : 159
Date solved : Tuesday, September 30, 2025 at 03:01:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} g \left (x \right ) y^{2}+f \left (x \right ) y y^{\prime }+a {y^{\prime }}^{2}+y y^{\prime \prime }&=0 \end{align*}

✗ Maple

ode:=g(x)*y(x)^2+f(x)*y(x)*diff(y(x),x)+a*diff(y(x),x)^2+y(x)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=g[x]*y[x]^2 + f[x]*y[x]*D[y[x],x] + a*D[y[x],x]^2 + 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") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*Derivative(y(x), x)**2 + f(x)*y(x)*Derivative(y(x), x) + g(x)*y(x)**2 + y(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((-4*a*g(x)*y(x) - 4*a*Derivative(y(x