13.2.12 problem 13

Internal problem ID [2310]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.2. Page 9
Problem number : 13
Date solved : Saturday, March 29, 2025 at 11:53:40 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.425 (sec). Leaf size: 49

ode:=diff(y(t),t)+y(t) = 1/(t^2+1); 
ic:=y(1) = 2; 
dsolve([ode,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: 0.04 (sec). Leaf size: 72

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

\[ y(t)\to \frac {1}{2} e^{-t-i} \left (-i e^{2 i} \operatorname {ExpIntegralEi}(t-i)+i \operatorname {ExpIntegralEi}(t+i)-i \operatorname {ExpIntegralEi}(1+i)+i e^{2 i} \operatorname {ExpIntegralEi}(1-i)+4 e^{1+i}\right ) \]

✓ Sympy. Time used: 7.073 (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)
 

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