Internal problem ID [11457]
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 2.3.1 Nonhomogeneous Equations:
Undetermined Coefficients. Exercises page 110
Problem number: 1(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x={\mathrm e}^{2 t} \cos \left (t \right )+t^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 52
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=exp(2*t)*cos(t)+t^2,x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +t^{2}-2 t +\frac {\left (5 \sin \left (t \right )+6 \cos \left (t \right )\right ) {\mathrm e}^{2 t}}{61} \]
✓ Solution by Mathematica
Time used: 2.836 (sec). Leaf size: 76
DSolve[x''[t]+x'[t]+x[t]==Exp[2*t]*Cos[t]+t^2,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to t^2-2 t+\frac {5}{61} e^{2 t} \sin (t)+\frac {6}{61} e^{2 t} \cos (t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right ) \]