7.23.3 problem 3

Internal problem ID [589]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 11:27:18 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-3 x \left (t \right )+4 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 2 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 33

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

\begin{align*} x \left (t \right ) &= -\frac {4 \,{\mathrm e}^{-2 t}}{5}+\frac {4 \,{\mathrm e}^{3 t}}{5} \\ y \left (t \right ) &= -\frac {2 \,{\mathrm e}^{-2 t}}{5}+\frac {12 \,{\mathrm e}^{3 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

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

\begin{align*} x(t)\to \frac {4}{5} e^{-2 t} \left (e^{5 t}-1\right ) \\ y(t)\to \frac {2}{5} e^{-2 t} \left (6 e^{5 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(3*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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