63.12.6 problem 1(f)

Internal problem ID [13168]
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(f)
Date solved : Tuesday, January 28, 2025 at 05:11:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-2 x^{\prime }+x&=\frac {{\mathrm e}^{t}}{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 29

DSolve[D[x[t],{t,2}]-2*D[x[t],t]+x[t]==1/(2*t)*Exp[t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{2} e^t (t \log (t)+(-1+2 c_2) t+2 c_1) \]