7.24.15 problem 25 and 34

Internal problem ID [615]
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 : 25 and 34
Date solved : Saturday, March 29, 2025 at 05:00:46 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )-3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 8\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.130 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = 4*x__1(t)-3*x__2(t), diff(x__2(t),t) = 6*x__1(t)-7*x__2(t)]; 
ic:=x__1(0) = 8x__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {16 \,{\mathrm e}^{-5 t}}{7}+\frac {72 \,{\mathrm e}^{2 t}}{7} \\ x_{2} \left (t \right ) &= -\frac {48 \,{\mathrm e}^{-5 t}}{7}+\frac {48 \,{\mathrm e}^{2 t}}{7} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

ode={D[x1[t],t]==4*x1[t]-3*x2[t],D[x2[t],t]==6*x1[t]-7*x2[t]}; 
ic={x1[0]==8,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {8}{7} e^{-5 t} \left (9 e^{7 t}-2\right ) \\ \text {x2}(t)\to \frac {48}{7} e^{-5 t} \left (e^{7 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__1(t) + 3*x__2(t) + Derivative(x__1(t), t),0),Eq(-6*x__1(t) + 7*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- 5 t}}{3} + \frac {3 C_{2} e^{2 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{2 t}\right ] \]