2.12 problem 13

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]

Solve \begin {gather*} \boxed {y+y^{\prime }-\frac {1}{t^{2}+1}=0} \end {gather*} With initial conditions \begin {align*} [y \relax (1) = 2] \end {align*}

Solution by Maple

Time used: 0.547 (sec). Leaf size: 64

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

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

Solution by Mathematica

Time used: 0.075 (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} (\text {Ei}(1-i)-\text {Ei}(t-i))-i (\text {Ei}(1+i)-\text {Ei}(t+i))+4 e^{1+i}\right ) \\ \end{align*}