60.7.71 problem 1680 (book 6.89)

Internal problem ID [11621]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1680 (book 6.89)
Date solved : Sunday, March 30, 2025 at 08:31:45 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.020 (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: 0.958 (sec). Leaf size: 54

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

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [c_1+\int _1^{K[3]}-\frac {1}{K[2]^2+1}dK[2]\right ]dK[3]+c_2 \]

✓ Sympy. Time used: 1.648 (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)
 

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