73.27.4 problem 38.4

Internal problem ID [15681]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.4
Date solved : Thursday, March 13, 2025 at 06:15:15 AM
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 )&=5 x \left (t \right )-2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 7\\ y \left (0\right ) = -7 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 33

ode:=[diff(x(t),t) = x(t)+2*y(t), diff(y(t),t) = 5*x(t)-2*y(t)]; 
ic:=x(0) = 7y(0) = -7; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{3 t}+4 \,{\mathrm e}^{-4 t} \\ y \left (t \right ) &= 3 \,{\mathrm e}^{3 t}-10 \,{\mathrm e}^{-4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 44

ode={D[x[t],t]==x[t]+2*y[t],D[y[t],t]==5*x[t]-2*y[t]}; 
ic={x[0]==8,y[0]==-7}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {2}{7} e^{-4 t} \left (13 e^{7 t}+15\right ) \\ y(t)\to \frac {1}{7} e^{-4 t} \left (26 e^{7 t}-75\right ) \\ \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 34

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(-5*x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{- 4 t}}{5} + C_{2} e^{3 t}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{3 t}\right ] \]