7.24.12 problem 22 and 31

Internal problem ID [612]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.3 (Matrices and linear systems). Problems at page 364
Problem number : 22 and 31
Date solved : Saturday, March 29, 2025 at 05:00:42 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*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 33

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)]; 
ic:=x__1(0) = 0x__2(0) = 5; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 37

ode={D[x1[t],t]==-3*x1[t]+2*x2[t],D[x2[t],t]==-3*x1[t]+4*x2[t]}; 
ic={x1[0]==0,x2[0]==5}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.118 (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)
 

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