9.3.1 problem problem 13

Internal problem ID [963]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.2, Matrices and Linear systems. Page 384
Problem number : problem 13
Date solved : Saturday, March 29, 2025 at 10:35:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=6 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-3 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.157 (sec). Leaf size: 27

ode:=[diff(x__1(t),t) = 6*x__1(t), diff(x__2(t),t) = -3*x__1(t)-x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= -\frac {3 c_2 \,{\mathrm e}^{6 t}}{7}+c_1 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 56

ode={D[ x1[t],t]==4*x1[t]+2*x2[t],D[ x2[t],t]==-3*x1[t]-x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t \left (c_1 \left (3 e^t-2\right )+2 c_2 \left (e^t-1\right )\right ) \\ \text {x2}(t)\to e^t \left (c_2 \left (3-2 e^t\right )-3 c_1 \left (e^t-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-6*x__1(t) + Derivative(x__1(t), t),0),Eq(3*x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {7 C_{1} e^{6 t}}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{6 t} + C_{2} e^{- t}\right ] \]