64.18.1 problem 1

Internal problem ID [13553]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.4. Exercises page 309
Problem number : 1
Date solved : Monday, March 31, 2025 at 08:01:10 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.111 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 63

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

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

✓ Sympy. Time used: 0.087 (sec). Leaf size: 29

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

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