76.7.4 problem 4

Internal problem ID [17406]
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 : 4
Date solved : Monday, March 31, 2025 at 04:13:02 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.126 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

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

\begin{align*} x(t)\to \frac {1}{5} e^{-3 t} \left (c_1 \left (4 e^{5 t}+1\right )+c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-3 t} \left (4 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 32

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

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