Internal problem ID [1660]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 1.2. Page 9
Problem number: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
\[ \boxed {y^{\prime }+y-\frac {1}{t^{2}+1}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.703 (sec). Leaf size: 65
dsolve([y(t)+diff(y(t),t) = 1/(t^2+1),y(1) = 2],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (-i \operatorname {Ei}_{1}\left (-1+i\right ) {\mathrm e}^{i}+i \operatorname {Ei}_{1}\left (-1-i\right ) {\mathrm e}^{-i}+i {\mathrm e}^{i} \operatorname {Ei}_{1}\left (-t +i\right )-i {\mathrm e}^{-i} \operatorname {Ei}_{1}\left (-t -i\right )+4 \,{\mathrm e}\right ) {\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 65
DSolve[{y[t]+y'[t] == 1/(t^2+1),y[1]==2},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} e^{-t-i} \left (i e^{2 i} (\operatorname {ExpIntegralEi}(1-i)-\operatorname {ExpIntegralEi}(t-i))-i (\operatorname {ExpIntegralEi}(1+i)-\operatorname {ExpIntegralEi}(t+i))+4 e^{1+i}\right ) \\ \end{align*}