19.1.2 problem 2

Internal problem ID [3516]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.4, page 36
Problem number : 2
Date solved : Sunday, March 30, 2025 at 01:45:33 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {y^{2}}{x^{2}+1} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 19

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

\begin{align*} y(x)\to -\frac {1}{\arctan (x)+c_1} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.170 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = - \frac {1}{C_{1} + \operatorname {atan}{\left (x \right )}} \]