50.29.13 problem 4(c)

Internal problem ID [8209]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 4(c)
Date solved : Monday, January 27, 2025 at 03:46:02 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+y \left (t \right )-t +3\\ y^{\prime }\left (t \right )&=-x \left (t \right )-5 y \left (t \right )+t +1 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 105

dsolve([diff(x(t),t)=-4*x(t)+y(t)-t+3,diff(y(t),t)=-x(t)-5*y(t)+t+1],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {4 t}{21}+\frac {39}{49} \\ y &= -\frac {{\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}+\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}-\frac {1}{147}+\frac {5 t}{21} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.459 (sec). Leaf size: 131

DSolve[{D[x[t],t]==-4*x[t]+y[t]-t+3,D[y[t],t]==-x[t]-5*y[t]+t+1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {4 t}{21}+c_1 e^{-9 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+\frac {(c_1+2 c_2) e^{-9 t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}+\frac {39}{49} \\ y(t)\to \frac {5 t}{21}+c_2 e^{-9 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )-\frac {(2 c_1+c_2) e^{-9 t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}-\frac {1}{147} \\ \end{align*}