6.2.27 problem 27

Internal problem ID [1563]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number : 27
Date solved : Tuesday, September 30, 2025 at 04:36:29 AM
CAS classification : [_linear]

\begin{align*} x y^{\prime }+3 y&=\frac {2}{x \left (x^{2}+1\right )} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 19

ode:=x*diff(y(x),x)+3*y(x) = 2/x/(x^2+1); 
ic:=[y(-1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\ln \left (x^{2}+1\right )-\ln \left (2\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 19

ode=x*D[y[x],x]+3*y[x]==2/(x*(1+x^2)); 
ic=y[-1]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\log \left (\frac {1}{2} \left (x^2+1\right )\right )}{x^3} \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + 3*y(x) - 2/(x*(x**2 + 1)),0) 
ics = {y(-1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\log {\left (x^{2} + 1 \right )} - \log {\left (2 \right )}}{x^{3}} \]