Internal problem ID [11489]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-x=\frac {1}{t}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
dsolve(diff(x(t),t$2)-x(t)=1/t,x(t), singsol=all)
\[ x \left (t \right ) = \frac {\operatorname {expIntegral}_{1}\left (-t \right ) {\mathrm e}^{-t}}{2}+c_{2} {\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_{1} -\frac {\operatorname {expIntegral}_{1}\left (t \right )}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 42
DSolve[x''[t]-x[t]==1/t,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{2} e^{-t} \left (e^{2 t} \operatorname {ExpIntegralEi}(-t)-\operatorname {ExpIntegralEi}(t)+2 \left (c_1 e^{2 t}+c_2\right )\right ) \]