67.27.5 problem 38.5

Internal problem ID [17048]
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 : 38.5
Date solved : Thursday, October 02, 2025 at 01:42:20 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )+y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=9 \\ \end{align*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 5*x(t)+4*y(t), diff(y(t),t) = 8*x(t)+y(t)]; 
ic:=[x(0) = 0, y(0) = 9]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{9 t} \\ y \left (t \right ) &= 6 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{9 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 36

ode={D[x[t],t]==5*x[t]+4*y[t],D[y[t],t]==8*x[t]+y[t]}; 
ic={x[0]==0,y[0]==9}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 3 e^{-3 t} \left (e^{12 t}-1\right )\\ y(t)&\to 3 e^{-3 t} \left (e^{12 t}+2\right ) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(-8*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{- 3 t}}{2} + C_{2} e^{9 t}, \ y{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{9 t}\right ] \]