52.10.19 problem 20

Internal problem ID [8413]
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 : 20
Date solved : Sunday, March 30, 2025 at 01:04:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-6 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: 0.112 (sec). Leaf size: 32

ode:=[diff(x(t),t) = -6*x(t)+5*y(t), diff(y(t),t) = -5*x(t)+4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (5 c_2 t +5 c_1 +c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[x[t],t]==-6*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 e^{-t} (-5 c_1 t+5 c_2 t+c_1) \\ y(t)\to e^{-t} (-5 c_1 t+5 c_2 t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*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)
 

\[ \left [ x{\left (t \right )} = - 5 C_{2} t e^{- t} - \left (5 C_{1} - C_{2}\right ) e^{- t}, \ y{\left (t \right )} = - 5 C_{1} e^{- t} - 5 C_{2} t e^{- t}\right ] \]