52.9.12 problem 12

Internal problem ID [8390]
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 8.1. Page 332
Problem number : 12
Date solved : Monday, January 27, 2025 at 03:57:24 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 43

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 51

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

\begin{align*} x(t)\to e^t (c_1 \cos (t)+(5 c_2-3 c_1) \sin (t)) \\ y(t)\to e^t (c_2 (3 \sin (t)+\cos (t))-2 c_1 \sin (t)) \\ \end{align*}