56.33.1 problem Ex 1

Internal problem ID [14269]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 57. Dependent variable absent. Page 132
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:27:29 AM
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: 0.020 (sec). Leaf size: 33

ode:=(x^2+1)*diff(diff(y(x),x),x)+1+diff(y(x),x)^2 = 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: 4.187 (sec). Leaf size: 33

ode=(1+x^2)*D[y[x],{x,2}]+1+(D[y[x],x])^2==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: 1.071 (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)
 

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