8.1.13 problem 13

Internal problem ID [2484]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.2. Linear equations. Excercises page 9
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 05:36:46 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }+y&=\frac {1}{t^{2}+1} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.419 (sec). Leaf size: 49

ode:=diff(y(t),t)+y(t) = 1/(t^2+1); 
ic:=[y(1) = 2]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-t} \left (i \left (-\operatorname {Ei}_{1}\left (-t -i\right )+\operatorname {Ei}_{1}\left (-1-i\right )\right ) {\mathrm e}^{-i}+i \left (\operatorname {Ei}_{1}\left (-t +i\right )-\operatorname {Ei}_{1}\left (-1+i\right )\right ) {\mathrm e}^{i}+4 \,{\mathrm e}\right )}{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 72

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

\begin{align*} y(t)&\to \frac {1}{2} e^{-t-i} \left (-i e^{2 i} \operatorname {ExpIntegralEi}(t-i)+i \operatorname {ExpIntegralEi}(t+i)-i \operatorname {ExpIntegralEi}(1+i)+i e^{2 i} \operatorname {ExpIntegralEi}(1-i)+4 e^{1+i}\right ) \end{align*}

✓ Sympy. Time used: 4.556 (sec). Leaf size: 63

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

\[ \int \frac {\left (t^{2} y{\left (t \right )} + y{\left (t \right )} - 1\right ) e^{t}}{t^{2} + 1}\, dt = \int \limits ^{1} \left (- \frac {e^{t}}{t^{2} + 1}\right )\, dt + \int \limits ^{1} \frac {y{\left (t \right )} e^{t}}{t^{2} + 1}\, dt + \int \limits ^{1} \frac {t^{2} y{\left (t \right )} e^{t}}{t^{2} + 1}\, dt \]