14.20.1 problem 1

Internal problem ID [2698]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 1
Date solved : Sunday, March 30, 2025 at 12:15:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=6 x \left (t \right )-3 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: 0.113 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}+c_2 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{4 t}}{3}+c_2 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 60

ode={D[x[t],t]==6*x[t]-3*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^{3 t} \left (c_1 \left (3 e^t-2\right )-3 c_2 \left (e^t-1\right )\right ) \\ y(t)\to e^{3 t} \left (2 c_1 \left (e^t-1\right )+c_2 \left (3-2 e^t\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-6*x(t) + 3*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)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{3 t} + \frac {3 C_{2} e^{4 t}}{2}, \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{4 t}\right ] \]