76.7.11 problem 11

Internal problem ID [17413]
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 : 11
Date solved : Monday, March 31, 2025 at 04:13:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-5 x \left (t \right )+4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.122 (sec). Leaf size: 34

ode:=[diff(x(t),t) = -2*x(t)+y(t), diff(y(t),t) = -5*x(t)+4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 5 c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 72

ode={D[x[t],t]==-2*x[t]+y[t],D[y[t],t]==-5*x[t]+4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) - y(t) + Derivative(x(t), t),0),Eq(5*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} + \frac {C_{2} e^{3 t}}{5}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{3 t}\right ] \]