23.4.128 problem 128

Internal problem ID [6430]
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 : 128
Date solved : Tuesday, September 30, 2025 at 02:56:38 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} {y^{\prime }}^{2}+y y^{\prime \prime }&=a^{2} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 43

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

\begin{align*} y &= \sqrt {a^{2} x^{2}-2 c_1 x +2 c_2} \\ y &= -\sqrt {a^{2} x^{2}-2 c_1 x +2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 17.806 (sec). Leaf size: 109

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

\begin{align*} y(x)&\to -\frac {\sqrt {a^4 (x+c_2){}^2-e^{2 c_1}}}{a}\\ y(x)&\to \frac {\sqrt {a^4 (x+c_2){}^2-e^{2 c_1}}}{a}\\ y(x)&\to -\frac {\sqrt {a^4 (x+c_2){}^2}}{a}\\ y(x)&\to \frac {\sqrt {a^4 (x+c_2){}^2}}{a} \end{align*}

✗ Sympy

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