70.7.2 problem 2

Internal problem ID [18762]
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 3.3 (Homogeneous linear systems with constant coefficients). Problems at page 165
Problem number : 2
Date solved : Thursday, October 02, 2025 at 03:30:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 34

ode:=[diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = 3*x(t)-4*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 60

ode={D[x[t],t]==x[t]-2*y[t],D[y[t],t]==3*x[t]-4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-2 t} \left (c_1 \left (3 e^t-2\right )-2 c_2 \left (e^t-1\right )\right )\\ y(t)&\to e^{-2 t} \left (3 c_1 \left (e^t-1\right )+c_2 \left (3-2 e^t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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