Internal problem ID [11461]
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(j).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x=5 \sin \left (2 t \right )+t \,{\mathrm e}^{t}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 50
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=5*sin(2*t)+t*exp(t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {10 \cos \left (2 t \right )}{13}-\frac {15 \sin \left (2 t \right )}{13}+\frac {{\mathrm e}^{t} \left (t -1\right )}{3} \]
✓ Solution by Mathematica
Time used: 5.753 (sec). Leaf size: 83
DSolve[x''[t]+x'[t]+x[t]==5*Sin[2*t]+t*Exp[t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{39} \left (-13 e^t+13 e^t t+30 \sin ^2(t)-30 \cos ^2(t)-90 \sin (t) \cos (t)+39 c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+39 c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]