Internal problem ID [10474]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-2 x^{\prime }+x-\frac {{\mathrm e}^{t}}{2 t}=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ x \left (t \right ) = c_{2} {\mathrm e}^{t}+{\mathrm e}^{t} t c_{1} +\frac {{\mathrm e}^{t} t \left (-1+\ln \left (t \right )\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 29
DSolve[x''[t]-2*x'[t]+x[t]==1/(2*t)*Exp[t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^t (t \log (t)+(-1+2 c_2) t+2 c_1) \\ \end{align*}