14.20.6 problem 6

Internal problem ID [2703]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:15:14 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 37

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

\begin{align*} x(t)\to -3 e^{-t} \left (e^{5 t}-1\right ) \\ y(t)\to e^{-t} \left (3 e^{5 t}+2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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