52.10.20 problem 21

Internal problem ID [8414]
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 : 21
Date solved : Sunday, March 30, 2025 at 01:04:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-3 x \left (t \right )+5 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 32

ode:=[diff(x(t),t) = -x(t)+3*y(t), diff(y(t),t) = -3*x(t)+5*y(t)]; 
dsolve(ode);
 

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

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

ode={D[x[t],t]==-x[t]+3*y[t],D[y[t],t]==-3*x[t]+5*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(3*x(t) - 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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