Internal problem ID [11498]
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(b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-x=t \,{\mathrm e}^{t}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve(diff(x(t),t$2)-x(t)=t*exp(t),x(t), singsol=all)
\[ x \left (t \right ) = c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+\frac {\left (t -1\right ) {\mathrm e}^{t} t}{4} \]
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 35
DSolve[x''[t]-x[t]==t*Exp[t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{8} e^t \left (2 t^2-2 t+1+8 c_1\right )+c_2 e^{-t} \]