7.23.4 problem 4

Internal problem ID [590]
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 : 4
Date solved : Tuesday, March 04, 2025 at 11:27:19 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 44

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

\begin{align*} x(t)\to \frac {1}{2} e^{-2 t} \left (3 e^{4 t}-1\right ) \\ y(t)\to \frac {1}{2} e^{-2 t} \left (3 e^{4 t}-5\right ) \\ \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + y(t) + Derivative(x(t), t),0),Eq(-5*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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