70.7.5 problem 5

Internal problem ID [18765]
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 : 5
Date solved : Thursday, October 02, 2025 at 03:30:41 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )-6 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 26

ode:=[diff(x(t),t) = 4*x(t)-3*y(t), diff(y(t),t) = 8*x(t)-6*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 59

ode={D[x[t],t]==4*x[t]-3*y[t],D[y[t],t]==8*x[t]-6*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 \left (3-2 e^{-2 t}\right )+\frac {3}{2} c_2 \left (e^{-2 t}-1\right )\\ y(t)&\to c_1 \left (4-4 e^{-2 t}\right )+c_2 \left (3 e^{-2 t}-2\right ) \end{align*}

✓ Sympy. Time used: 0.045 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(-8*x(t) + 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {3 C_{1}}{4} + \frac {C_{2} e^{- 2 t}}{2}, \ y{\left (t \right )} = C_{1} + C_{2} e^{- 2 t}\right ] \]