70.9.1 problem 1

Internal problem ID [18798]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 1
Date solved : Thursday, October 02, 2025 at 03:31:00 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 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: 0.109 (sec). Leaf size: 30

ode:=[diff(x(t),t) = 3*x(t)-4*y(t), diff(y(t),t) = x(t)-y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 41

ode={D[x[t],t]==3*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^t (2 c_1 t-4 c_2 t+c_1)\\ y(t)&\to e^t ((c_1-2 c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*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)
 

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