20.3.6 problem Problem 6

Internal problem ID [3615]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.6, First-Order Linear Differential Equations. page 59
Problem number : Problem 6
Date solved : Tuesday, March 04, 2025 at 04:54:57 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }+\frac {2 x y}{x^{2}+1}&=\frac {4}{\left (x^{2}+1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

ode:=diff(y(x),x)+2*x/(x^2+1)*y(x) = 4/(x^2+1)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {4 \arctan \left (x \right )+c_{1}}{x^{2}+1} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 20

ode=D[y[x],x]+2*x/(1+x^2)*y[x]==4/(1+x^2)^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {4 \arctan (x)+c_1}{x^2+1} \]

✓ Sympy. Time used: 0.307 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*y(x)/(x**2 + 1) + Derivative(y(x), x) - 4/(x**2 + 1)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} - 2 i \log {\left (x - i \right )} + 2 i \log {\left (x + i \right )}}{x^{2} + 1} \]