64.17.2 problem 2

Internal problem ID [13551]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.3. Exercises page 299
Problem number : 2
Date solved : Monday, March 31, 2025 at 08:01:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 8 \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 5*x(t)+3*y(t), diff(y(t),t) = 4*x(t)+y(t)]; 
ic:=x(0) = 0y(0) = 8; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{-t}+3 \,{\mathrm e}^{7 t} \\ y \left (t \right ) &= 6 \,{\mathrm e}^{-t}+2 \,{\mathrm e}^{7 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[x[t],t]==5*x[t]+3*y[t],D[y[t],t]==4*x[t]+y[t]}; 
ic={x[0]==0,y[0]==8}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 3 e^{-t} \left (e^{8 t}-1\right ) \\ y(t)\to 2 e^{-t} \left (e^{8 t}+3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{- t}}{2} + \frac {3 C_{2} e^{7 t}}{2}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{7 t}\right ] \]