81.13.7 problem 17-7

Internal problem ID [21681]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 17. Reduction of Order. Page 420
Problem number : 17-7
Date solved : Thursday, October 02, 2025 at 07:59:53 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 y^{\prime \prime }+{y^{\prime }}^{2}&=1 \end{align*}

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

ode:=y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2 = 1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.37 (sec). Leaf size: 79

ode=y[x]*D[y[x],{x,2}]+D[y[x],x]^2==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-y(x)*Derivative(y(x), (x, 2)) + 1) + Derivative(y(x), x) cannot be solved by the factorable group method