23.4.166 problem 166

Internal problem ID [6468]
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 : 166
Date solved : Tuesday, September 30, 2025 at 03:01:36 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 35

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

\begin{align*} y &= -a \\ y &= \left (\frac {1}{\left (b +1\right ) \left (c_1 x +c_2 \right )}\right )^{-\frac {1}{b +1}}-a \\ \end{align*}

✓ Mathematica. Time used: 3.807 (sec). Leaf size: 25

ode=b*D[y[x],x]^2 + (a + y[x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -a+((b+1) c_1 (x+c_2)){}^{\frac {1}{b+1}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-(a + y(x))*Derivative(y(x), (x, 2))/b) + Derivative(y(x),