30.6.40 problem 41

Internal problem ID [7575]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problems. page 79
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 04:54:23 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (2\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.225 (sec). Leaf size: 59

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

\[ y = \frac {{\mathrm e}^{-t} \left (i \left (\operatorname {Ei}_{1}\left (-t +i\right )-\operatorname {Ei}_{1}\left (-2+i\right )\right ) {\mathrm e}^{i}+i \left (-\operatorname {Ei}_{1}\left (-t -i\right )+\operatorname {Ei}_{1}\left (-2-i\right )\right ) {\mathrm e}^{-i}+6 \,{\mathrm e}^{2}\right )}{2} \]

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 72

ode=D[y[t],t]==1/(1+t^2) - y[t]; 
ic={y[2]==3}; 
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}(2+i)+i e^{2 i} \operatorname {ExpIntegralEi}(2-i)+6 e^{2+i}\right ) \end{align*}

✓ Sympy. Time used: 3.073 (sec). Leaf size: 92

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

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