50.3.25 problem 25

Internal problem ID [10169]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 07:10:57 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}&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 14

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

\[ y = \int \frac {1}{\arctan \left (x \right )+c_1}d x +c_2 \]

✓ Mathematica. Time used: 3.161 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to \int _1^x-\frac {1}{c_1+\int _1^{K[2]}-\frac {1}{K[1]^2+1}dK[1]}dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 0.905 (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)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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