12.2.15 problem 15

Internal problem ID [1551]
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 : 15
Date solved : Saturday, March 29, 2025 at 10:58:40 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

\[ y = \frac {\sqrt {\pi }\, \operatorname {erf}\left (x \right )+2 c_1}{2 x^{2}+2} \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 28

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

\[ y(x)\to \frac {\sqrt {\pi } \text {erf}(x)+2 c_1}{2 x^2+2} \]

✓ Sympy. Time used: 11.795 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \frac {C_{1} + \frac {\sqrt {\pi } \operatorname {erf}{\left (x \right )}}{2}}{x^{2} + 1} \]