63.12.3 problem 1(c)

Internal problem ID [13165]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.2 Variation of parameters. Exercises page 124
Problem number : 1(c)
Date solved : Tuesday, January 28, 2025 at 05:11:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-x&=\frac {1}{t} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 32

dsolve(diff(x(t),t$2)-x(t)=1/t,x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\operatorname {Ei}_{1}\left (-t \right ) {\mathrm e}^{-t}}{2}+c_{2} {\mathrm e}^{-t}+\left (c_{1} -\frac {\operatorname {Ei}_{1}\left (t \right )}{2}\right ) {\mathrm e}^{t} \]

✓ Solution by Mathematica

Time used: 0.058 (sec). Leaf size: 68

DSolve[D[x[t],{t,2}]-x[t]==1/t,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{-t} \left (e^{2 t} \int _1^t\frac {e^{-K[1]}}{2 K[1]}dK[1]+\int _1^t-\frac {e^{K[2]}}{2 K[2]}dK[2]+c_1 e^{2 t}+c_2\right ) \]