68.22.8 problem 8

Internal problem ID [17942]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 8
Date solved : Thursday, October 02, 2025 at 02:30:09 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=8 x-y \left (t \right )\\ y^{\prime }\left (t \right )&=x+6 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.104 (sec). Leaf size: 30

ode:=[diff(x(t),t) = 8*x(t)-y(t), diff(y(t),t) = x(t)+6*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{7 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= {\mathrm e}^{7 t} \left (c_2 t +c_1 -c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 44

ode={D[x[t],t]==8*x[t]-y[t],D[y[t],t]==x[t]+6*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{7 t} (c_1 (t+1)-c_2 t)\\ y(t)&\to e^{7 t} ((c_1-c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-8*x(t) + y(t) + Derivative(x(t), t),0),Eq(-x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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