12.2.26 problem 26

Internal problem ID [1562]
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 : 26
Date solved : Saturday, March 29, 2025 at 10:59:13 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 17

ode:=(x^2+1)*diff(y(x),x)+4*x*y(x) = 2/(x^2+1); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 18

ode=(1+x^2)*D[y[x],x]+4*x*y[x]==2/(1+x^2); 
ic=y[0]==1; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {2 x+1}{\left (x^2+1\right )^2} \]

✓ Sympy. Time used: 0.299 (sec). Leaf size: 17

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

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