Internal problem ID [10452]
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: 2(h).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+2 x-\sin \left (2 t \right ) t=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 51
dsolve(diff(x(t),t$2)+diff(x(t),t)+2*x(t)=t*sin(2*t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1} +\frac {\left (-2 t -1\right ) \cos \left (2 t \right )}{8}-\frac {\sin \left (2 t \right ) \left (t -2\right )}{4} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 67
DSolve[x''[t]+x'[t]+2*x[t]==t*Sin[2*t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {1}{4} (t-2) \sin (2 t)-\frac {1}{8} (2 t+1) \cos (2 t)+e^{-t/2} \left (c_2 \cos \left (\frac {\sqrt {7} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \\ \end{align*}