52.10.3 problem 3

Internal problem ID [8397]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 3
Date solved : Monday, January 27, 2025 at 03:57:30 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+2 y\\ y^{\prime }&=-\frac {5 x \left (t \right )}{2}+2 y \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 31

dsolve([diff(x(t),t)=-4*x(t)+2*y(t),diff(y(t),t)=-5/2*x(t)+2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-3 t} \\ y &= \frac {5 c_{1} {\mathrm e}^{t}}{2}+\frac {c_{2} {\mathrm e}^{-3 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 149

DSolve[{D[x[t],t]==-4*x[t]+2*y[t],D[y[t],t]==5/2*x[t]+2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{28} e^{-\left (\left (1+\sqrt {14}\right ) t\right )} \left (c_1 \left (\left (14-3 \sqrt {14}\right ) e^{2 \sqrt {14} t}+14+3 \sqrt {14}\right )+2 \sqrt {14} c_2 \left (e^{2 \sqrt {14} t}-1\right )\right ) \\ y(t)\to \frac {1}{56} e^{-\left (\left (1+\sqrt {14}\right ) t\right )} \left (5 \sqrt {14} c_1 \left (e^{2 \sqrt {14} t}-1\right )+2 c_2 \left (\left (14+3 \sqrt {14}\right ) e^{2 \sqrt {14} t}+14-3 \sqrt {14}\right )\right ) \\ \end{align*}