73.28.3 problem 39.1 (c)

Internal problem ID [15699]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 39. Critical points, Direction fields and trajectories. Additional Exercises. page 815
Problem number : 39.1 (c)
Date solved : Thursday, March 13, 2025 at 06:15:38 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 )&=6 x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 26

ode:=[diff(x(t),t) = 3*x(t)+y(t), diff(y(t),t) = 6*x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{5 t} \\ y \left (t \right ) &= 2 c_{2} {\mathrm e}^{5 t}-3 c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 63

ode={D[x[t],t]==3*x[t]+y[t],D[y[t],t]==6*x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} \left (c_1 \left (3 e^{5 t}+2\right )+c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} \left (6 c_1 \left (e^{5 t}-1\right )+c_2 \left (2 e^{5 t}+3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 24

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(-6*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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