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 : Sunday, March 30, 2025 at 01:04:38 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.126 (sec). Leaf size: 49

ode:=[diff(x(t),t) = 4*x(t)-5*y(t), diff(y(t),t) = 5*x(t)-4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (3 t \right )+c_2 \cos \left (3 t \right ) \\ y \left (t \right ) &= -\frac {3 c_1 \cos \left (3 t \right )}{5}+\frac {3 c_2 \sin \left (3 t \right )}{5}+\frac {4 c_1 \sin \left (3 t \right )}{5}+\frac {4 c_2 \cos \left (3 t \right )}{5} \\ \end{align*}

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

ode={D[x[t],t]==4*x[t]-5*y[t],D[y[t],t]==5*x[t]-4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {3 C_{1}}{5} - \frac {4 C_{2}}{5}\right ) \cos {\left (3 t \right )} - \left (\frac {4 C_{1}}{5} + \frac {3 C_{2}}{5}\right ) \sin {\left (3 t \right )}, \ y{\left (t \right )} = - C_{1} \sin {\left (3 t \right )} + C_{2} \cos {\left (3 t \right )}\right ] \]