65.17.3 problem 30.1 (iii)

Internal problem ID [13765]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 30, A repeated real eigenvalue. Exercises page 299
Problem number : 30.1 (iii)
Date solved : Wednesday, March 05, 2025 at 10:15:08 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 30

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} -c_{2} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 44

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

\begin{align*} x(t)\to e^{-4 t} (c_1 (t+1)-c_2 t) \\ y(t)\to e^{-4 t} ((c_1-c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 36

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

\[ \left [ x{\left (t \right )} = C_{2} t e^{- 4 t} + \left (C_{1} + C_{2}\right ) e^{- 4 t}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} t e^{- 4 t}\right ] \]