Internal problem ID [10445]
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(L).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x+6-2 \,{\mathrm e}^{2 t} \sin \left (t \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 47
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=-6+2*exp(2*t)*sin(t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -6+\frac {2 \left (-5 \cos \left (t \right )+6 \sin \left (t \right )\right ) {\mathrm e}^{2 t}}{61} \]
✓ Solution by Mathematica
Time used: 0.56 (sec). Leaf size: 62
DSolve[x''[t]+x'[t]+x[t]==-6+2*Exp[2*t]*Sin[t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {2}{61} e^{2 t} (5 \cos (t)-6 \sin (t))+e^{-t/2} \left (c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right )-6 \\ \end{align*}