63.5.21 problem 6

Internal problem ID [13095]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 04:52:08 AM
CAS classification : [_linear]

\begin{align*} x^{\prime }+\frac {{\mathrm e}^{-t} x}{t}&=t \end{align*}

With initial conditions

\begin{align*} x \left (1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.461 (sec). Leaf size: 23

dsolve([diff(x(t),t)+exp(-t)/t*x(t)=t,x(1) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \left (\int _{1}^{t}\textit {\_z1} \,{\mathrm e}^{-\operatorname {Ei}_{1}\left (\textit {\_z1} \right )}d \textit {\_z1} \right ) {\mathrm e}^{\operatorname {Ei}_{1}\left (t \right )} \]

✓ Solution by Mathematica

Time used: 0.101 (sec). Leaf size: 61

DSolve[{D[x[t],t]+Exp[-t]/t*x[t]==t,{x[1]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \exp \left (\int _1^t-\frac {e^{-K[1]}}{K[1]}dK[1]\right ) \int _1^t\exp \left (-\int _1^{K[2]}-\frac {e^{-K[1]}}{K[1]}dK[1]\right ) K[2]dK[2] \]