52.10.34 problem 37

Internal problem ID [8428]
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 : 37
Date solved : Monday, January 27, 2025 at 04:00:24 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= \sin \left (3 t \right ) c_{1} +\cos \left (3 t \right ) c_{2} \\ y &= -\frac {3 \cos \left (3 t \right ) c_{1}}{5}+\frac {3 \sin \left (3 t \right ) c_{2}}{5}+\frac {4 \sin \left (3 t \right ) c_{1}}{5}+\frac {4 \cos \left (3 t \right ) c_{2}}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 58

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

\begin{align*} x(t)\to c_1 \cos (3 t)+\frac {1}{3} (4 c_1-5 c_2) \sin (3 t) \\ y(t)\to c_2 \cos (3 t)+\frac {1}{3} (5 c_1-4 c_2) \sin (3 t) \\ \end{align*}