9.3.2 problem problem 14

Internal problem ID [964]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.2, Matrices and Linear systems. Page 384
Problem number : problem 14
Date solved : Saturday, March 29, 2025 at 10:35:29 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-3 x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -3*x__1(t)+2*x__2(t), diff(x__2(t),t) = -3*x__1(t)+4*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 73

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

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-2 t} \left (2 c_2 \left (e^{5 t}-1\right )-c_1 \left (e^{5 t}-6\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-2 t} \left (c_2 \left (6 e^{5 t}-1\right )-3 c_1 \left (e^{5 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) - 2*x__2(t) + Derivative(x__1(t), t),0),Eq(3*x__1(t) - 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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