65.15.2 problem 28.2 (ii)

Internal problem ID [13755]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 28, Distinct real eigenvalues. Exercises page 282
Problem number : 28.2 (ii)
Date solved : Wednesday, March 05, 2025 at 10:14:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-5 x \left (t \right )-3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 27

ode:=[diff(x(t),t) = 2*x(t), diff(y(t),t) = -5*x(t)-3*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[x[t],t]==2*x[t],D[y[t],t]==-5*x[t]-3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.077 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + Derivative(x(t), t),0),Eq(5*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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