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 : Monday, January 27, 2025 at 05:45:19 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*}

✓ Solution by Maple

Time used: 0.375 (sec). Leaf size: 51

dsolve([y(t)+diff(y(t),t) = 1/(t^2+1),y(1) = 2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 72

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

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