59.17.1 problem 30.1 (i)

Internal problem ID [15123]
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 (i)
Date solved : Thursday, October 02, 2025 at 10:03:16 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (2 c_2 t +2 c_1 -c_2 \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 45

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

\begin{align*} x(t)&\to e^{3 t} (2 c_1 t-4 c_2 t+c_1)\\ y(t)&\to e^{3 t} ((c_1-2 c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.060 (sec). Leaf size: 39

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

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