52.9.11 problem 11

Internal problem ID [8389]
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 8.1. Page 332
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:00:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.109 (sec). Leaf size: 31

ode:=[diff(x(t),t) = 3*x(t)-4*y(t), diff(y(t),t) = 4*x(t)-7*y(t)]; 
dsolve(ode);
 

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

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

ode={D[x[t],t]==3*x[t]-4*y[t],D[y[t],t]==4*x[t]-7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-5 t} \left (c_1 \left (4 e^{6 t}-1\right )-2 c_2 \left (e^{6 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-5 t} \left (2 c_1 \left (e^{6 t}-1\right )-c_2 \left (e^{6 t}-4\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 4*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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