63.9.12 problem 1(L)

Internal problem ID [13139]
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(L)
Date solved : Tuesday, January 28, 2025 at 05:05:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=-6+2 \,{\mathrm e}^{2 t} \sin \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 47

dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=-6+2*exp(2*t)*sin(t),x(t), singsol=all)
 

\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -6+\frac {2 \left (-5 \cos \left (t \right )+6 \sin \left (t \right )\right ) {\mathrm e}^{2 t}}{61} \]

✓ Solution by Mathematica

Time used: 0.339 (sec). Leaf size: 158

DSolve[D[x[t],{t,2}]+D[x[t],t]+x[t]==-6+2*Exp[2*t]*Sin[t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{-t/2} \left (\cos \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t-\frac {4 e^{\frac {K[2]}{2}} \left (e^{2 K[2]} \sin (K[2])-3\right ) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )}{\sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t\frac {4 e^{\frac {K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) \left (e^{2 K[1]} \sin (K[1])-3\right )}{\sqrt {3}}dK[1]+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]