70.7.16 problem 16

Internal problem ID [18776]
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 : 16
Date solved : Thursday, October 02, 2025 at 03:30:46 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*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 33

ode:=[diff(x(t),t) = -2*x(t)+y(t), diff(y(t),t) = -5*x(t)+4*y(t)]; 
ic:=[x(0) = 1, y(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{3 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{3 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 42

ode={D[x[t],t]==-2*x[t]+y[t],D[y[t],t]==-5*x[t]+4*y[t]}; 
ic={x[0]==1,y[0]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} e^{-t} \left (e^{4 t}+1\right )\\ y(t)&\to \frac {1}{2} e^{-t} \left (5 e^{4 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.047 (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)
 

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