63.9.10 problem 1(j)

Internal problem ID [13137]
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)
Date solved : Tuesday, January 28, 2025 at 05:02:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=5 \sin \left (2 t \right )+t \,{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.006 (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)
 

\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {10 \cos \left (2 t \right )}{13}-\frac {15 \sin \left (2 t \right )}{13}+\frac {\left (t -1\right ) {\mathrm e}^{t}}{3} \]

✓ Solution by Mathematica

Time used: 1.319 (sec). Leaf size: 164

DSolve[D[x[t],{t,2}]+D[x[t],t]+x[t]==5*Sin[2*t]+t*Exp[t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{-t/2} \left (\cos \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t-\frac {2 e^{\frac {K[2]}{2}} \left (e^{K[2]} K[2]+5 \sin (2 K[2])\right ) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )}{\sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t\frac {2 e^{\frac {K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) \left (e^{K[1]} K[1]+5 \sin (2 K[1])\right )}{\sqrt {3}}dK[1]+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]