23.4.49 problem 49

Internal problem ID [6351]
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 : 49
Date solved : Tuesday, September 30, 2025 at 02:52:18 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=h[y[x]] + f[y[x]]*D[y[x],x] + g[y[x]]*D[y[x],x]^2 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(f(y(x))*Derivative(y(x), x) + g(y(x))*Derivative(y(x), x)**2 + h(y(x)) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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