82.8.2 problem 36-3

Internal problem ID [21873]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-3
Date solved : Thursday, October 02, 2025 at 08:03:12 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

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

\[ y = \frac {\ln \left (c_1 x -1\right ) c_1^{2}+c_2 \,c_1^{2}+c_1 x +\ln \left (c_1 x -1\right )}{c_1^{2}} \]

✓ Mathematica. Time used: 4.042 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to -x \cot (c_1)+\csc ^2(c_1) \log (-x \sin (c_1)-\cos (c_1))+c_2 \end{align*}

✓ Sympy. Time used: 1.087 (sec). Leaf size: 24

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

\[ \left [ y{\left (x \right )} = C_{1} + \int \tan {\left (C_{2} - \operatorname {atan}{\left (x \right )} \right )}\, dx, \ y{\left (x \right )} = C_{1} + \int \tan {\left (C_{2} - \operatorname {atan}{\left (x \right )} \right )}\, dx\right ] \]