81.8.4 problem 9-4

Internal problem ID [21587]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 9. Clairaut Equation. Page 133.
Problem number : 9-4
Date solved : Thursday, October 02, 2025 at 07:58:41 PM
CAS classification : [_quadrature]

\begin{align*} \left (x^{2}-2 x \right ) \left (1+{y^{\prime }}^{2}\right )+1&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 38

ode:=(x^2-2*x)*(1+diff(y(x),x)^2)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x \left (x -2\right )}{\sqrt {-x \left (x -2\right )}}+c_1 \\ y &= -\frac {x \left (x -2\right )}{\sqrt {-x \left (x -2\right )}}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 65

ode=(x^2-2*x)*( D[y[x],x]^2+1)+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {(x-2) x} (x-1)}{\sqrt {-(x-1)^2}}+c_1\\ y(x)&\to \frac {\sqrt {-(x-1)^2} \sqrt {(x-2) x}}{x-1}+c_1 \end{align*}

✓ Sympy. Time used: 0.920 (sec). Leaf size: 66

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

\[ \left [ y{\left (x \right )} = C_{1} - \int x \sqrt {- \frac {1}{x \left (x - 2\right )}}\, dx + \int \sqrt {- \frac {1}{x \left (x - 2\right )}}\, dx, \ y{\left (x \right )} = C_{1} + x^{2} \sqrt {- \frac {1}{x \left (x - 2\right )}} - 2 x \sqrt {- \frac {1}{x \left (x - 2\right )}}\right ] \]