7.24.14 problem 24 and 33

Internal problem ID [614]
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 : 24 and 33
Date solved : Saturday, March 29, 2025 at 05:00:45 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 11\\ x_{2} \left (0\right ) = -7 \end{align*}

✓ Maple. Time used: 0.141 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = 4*x__1(t)+x__2(t), diff(x__2(t),t) = -2*x__1(t)+x__2(t)]; 
ic:=x__1(0) = 11x__2(0) = -7; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 15 \,{\mathrm e}^{3 t}-4 \,{\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= -15 \,{\mathrm e}^{3 t}+8 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 34

ode={D[x1[t],t]==4*x1[t]+x2[t],D[x2[t],t]==-2*x1[t]+x2[t]}; 
ic={x1[0]==11,x2[0]==-7}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} \left (15 e^t-4\right ) \\ \text {x2}(t)\to e^{2 t} \left (8-15 e^t\right ) \\ \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 34

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

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