20.2.12 problem Problem 12

Internal problem ID [3604]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.4, Separable Differential Equations. page 43
Problem number : Problem 12
Date solved : Sunday, March 30, 2025 at 01:54:12 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 15

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

\[ y = \frac {1-x}{x +1} \]

✓ Mathematica. Time used: 0.269 (sec). Leaf size: 14

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

\[ y(x)\to \cot \left (\arctan (x)+\frac {\pi }{4}\right ) \]

✓ Sympy. Time used: 0.284 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \cot {\left (\operatorname {atan}{\left (x \right )} + \frac {\pi }{4} \right )} \]