52.10.3 problem 3

Internal problem ID [8397]
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 : 3
Date solved : Wednesday, March 05, 2025 at 05:44:36 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-\frac {5 x \left (t \right )}{2}+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 31

ode:=[diff(x(t),t) = -4*x(t)+2*y(t), diff(y(t),t) = -5/2*x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-3 t} \\ y &= \frac {5 c_{1} {\mathrm e}^{t}}{2}+\frac {c_{2} {\mathrm e}^{-3 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 149

ode={D[x[t],t]==-4*x[t]+2*y[t],D[y[t],t]==5/2*x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{28} e^{-\left (\left (1+\sqrt {14}\right ) t\right )} \left (c_1 \left (\left (14-3 \sqrt {14}\right ) e^{2 \sqrt {14} t}+14+3 \sqrt {14}\right )+2 \sqrt {14} c_2 \left (e^{2 \sqrt {14} t}-1\right )\right ) \\ y(t)\to \frac {1}{56} e^{-\left (\left (1+\sqrt {14}\right ) t\right )} \left (5 \sqrt {14} c_1 \left (e^{2 \sqrt {14} t}-1\right )+2 c_2 \left (\left (14+3 \sqrt {14}\right ) e^{2 \sqrt {14} t}+14-3 \sqrt {14}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(4*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(5*x(t)/2 - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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