Internal problem ID [10443]
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]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x-5 \sin \left (2 t \right )-{\mathrm e}^{t} t=0} \]
✓ 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: 1.831 (sec). Leaf size: 70
DSolve[x''[t]+x'[t]+x[t]==5*Sin[2*t]+t*Exp[t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{3} e^t (t-1)-\frac {5}{13} (3 \sin (2 t)+2 \cos (2 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 ) \\ \end{align*}