50.29.5 problem 3(a)

Internal problem ID [8201]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 3(a)
Date solved : Wednesday, March 05, 2025 at 05:31:59 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 40

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

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

✓ Sympy. Time used: 0.086 (sec). Leaf size: 37

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

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