78.29.1 problem 1 (a)

Internal problem ID [18401]
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 60. Critical Points and Stability for Linear Systems. Problems at page 539
Problem number : 1 (a)
Date solved : Thursday, March 13, 2025 at 11:55:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 65

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

\begin{align*} x(t)\to c_1 e^{2 t} \\ y(t)\to c_2 e^{3 t} \\ x(t)\to c_1 e^{2 t} \\ y(t)\to 0 \\ x(t)\to 0 \\ y(t)\to c_2 e^{3 t} \\ x(t)\to 0 \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + Derivative(x(t), t),0),Eq(-3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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