23.4.164 problem 164

Internal problem ID [6466]
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 : 164
Date solved : Tuesday, September 30, 2025 at 03:01:32 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 20

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

\begin{align*} y &= -a \\ y &= {\mathrm e}^{c_1 x} c_2 -a \\ \end{align*}

✓ Mathematica. Time used: 0.367 (sec). Leaf size: 18

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq((a + 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((a + y(x))*Derivative(y(x), (x, 2))) + Derivative(y(x), x)