Internal problem ID [11491]
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(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x=\frac {1}{1+t}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
dsolve(diff(x(t),t$2)+x(t)=1/(1+t),x(t), singsol=all)
\[ x \left (t \right ) = \sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} -\operatorname {Si}\left (t +1\right ) \cos \left (t +1\right )+\operatorname {Ci}\left (t +1\right ) \sin \left (t +1\right ) \]
✓ Solution by Mathematica
Time used: 0.125 (sec). Leaf size: 35
DSolve[x''[t]+x[t]==1/(1+t),x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \operatorname {CosIntegral}(t+1) \sin (t+1)-\text {Si}(t+1) \cos (t+1)+c_1 \cos (t)+c_2 \sin (t) \]