Internal problem ID [11735]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 18, The variation of constants formula. Exercises page 168
Problem number: 18.1 (ii).
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: 33
dsolve(diff(x(t),t$2)-x(t)=1/t,x(t), singsol=all)
\[ x \left (t \right ) = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}-\frac {\operatorname {Ei}_{1}\left (t \right ) {\mathrm e}^{t}}{2}+\frac {\operatorname {Ei}_{1}\left (-t \right ) {\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 42
DSolve[x''[t]-x[t]==1/t,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{2} e^{-t} \left (e^{2 t} \operatorname {ExpIntegralEi}(-t)-\operatorname {ExpIntegralEi}(t)+2 \left (c_1 e^{2 t}+c_2\right )\right ) \]