63.9.19 problem 2(h)

Internal problem ID [13146]
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)
Date solved : Tuesday, January 28, 2025 at 05:06:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+2 x&=t \sin \left (2 t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (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: 0.036 (sec). Leaf size: 72

DSolve[D[x[t],{t,2}]+D[x[t],t]+2*x[t]==t*Sin[2*t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -\frac {1}{4} (t-2) \sin (2 t)-\frac {1}{8} (2 t+1) \cos (2 t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {7} t}{2}\right ) \]