81.8.6 problem 9-6

Internal problem ID [21589]
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 : 9-6
Date solved : Thursday, October 02, 2025 at 07:58:44 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 35

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

\begin{align*} y &= \arctan \left (\sqrt {{\mathrm e}^{-4+2 x}-1}\right )+c_1 \\ y &= -\arctan \left (\sqrt {{\mathrm e}^{-4+2 x}-1}\right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 1.067 (sec). Leaf size: 44

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

\[ \left [ y{\left (x \right )} = C_{1} - e^{2} \int \sqrt {\frac {1}{e^{2 x} - e^{4}}}\, dx, \ y{\left (x \right )} = C_{1} + e^{2} \int \sqrt {\frac {1}{e^{2 x} - e^{4}}}\, dx\right ] \]