73.28.5 problem 39.2

Internal problem ID [15709]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 39. Critical points, Direction fields and trajectories. Additional Exercises. page 815
Problem number : 39.2
Date solved : Monday, March 31, 2025 at 01:45:42 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 )&=2 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 68

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

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

✓ Sympy. Time used: 0.107 (sec). Leaf size: 27

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(-2*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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