52.10.6 problem 6

Internal problem ID [8400]
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 : 6
Date solved : Sunday, March 30, 2025 at 01:00:49 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-6 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-3 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 26

ode:=[diff(x(t),t) = -6*x(t)+2*y(t), diff(y(t),t) = -3*x(t)+y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-5 t} \\ y \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{-5 t}}{2}+3 c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 73

ode={D[x[t],t]==-6*x[t]+2*y[t],D[y[t],t]==-3*x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-5 t} \left (2 c_2 \left (e^{5 t}-1\right )-c_1 \left (e^{5 t}-6\right )\right ) \\ y(t)\to \frac {1}{5} e^{-5 t} \left (c_2 \left (6 e^{5 t}-1\right )-3 c_1 \left (e^{5 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.077 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(3*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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