63.9.11 problem 1(k)

Internal problem ID [13138]
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(k)
Date solved : Tuesday, January 28, 2025 at 05:03:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=t^{3}+1-4 t \cos \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 52

dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=t^3+1-4*t*cos(t),x(t), singsol=all)
 

\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\left (-4 t +8\right ) \sin \left (t \right )+t^{3}-3 t^{2}-4 \cos \left (t \right )+7 \]

✓ Solution by Mathematica

Time used: 0.550 (sec). Leaf size: 160

DSolve[D[x[t],{t,2}]+D[x[t],t]+x[t]==t^3+1-4*t*Cos[t],x[t],t,IncludeSingularSolutions -> True]
 

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