78.28.7 problem 1 (g)

Internal problem ID [18398]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 56. Homogeneous Linear Systems with Constant Coefficients. Problems at page 505
Problem number : 1 (g)
Date solved : Thursday, March 13, 2025 at 11:55:35 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=7 x \left (t \right )+6 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+6 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 7*x(t)+6*y(t), diff(y(t),t) = 2*x(t)+6*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{10 t} \\ y &= -\frac {2 c_{1} {\mathrm e}^{3 t}}{3}+\frac {c_{2} {\mathrm e}^{10 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

ode={D[x[t],t]==7*x[t]+6*y[t],D[y[t],t]==2*x[t]+6*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{7} e^{3 t} \left (c_1 \left (4 e^{7 t}+3\right )+6 c_2 \left (e^{7 t}-1\right )\right ) \\ y(t)\to \frac {1}{7} e^{3 t} \left (2 c_1 \left (e^{7 t}-1\right )+c_2 \left (3 e^{7 t}+4\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-7*x(t) - 6*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {3 C_{1} e^{3 t}}{2} + 2 C_{2} e^{10 t}, \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{10 t}\right ] \]