7.24.16 problem 26 and 35

Internal problem ID [616]
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 : 26 and 35
Date solved : Saturday, March 29, 2025 at 05:00:47 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 )&=-x_{1} \left (t \right )+3 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 4 \end{align*}

✓ Maple. Time used: 0.178 (sec). Leaf size: 51

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t), diff(x__2(t),t) = -x__1(t)+3*x__2(t)-2*x__3(t), diff(x__3(t),t) = -x__2(t)+3*x__3(t)]; 
ic:=x__1(0) = 0x__2(0) = 0x__3(0) = 4; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{5 t}-4 \,{\mathrm e}^{3 t}+2 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -2 \,{\mathrm e}^{5 t}+2 \,{\mathrm e}^{t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{5 t}+2 \,{\mathrm e}^{3 t}+{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 50

ode={D[x1[t],t]==3*x1[t]-2*x2[t],D[x2[t],t]==-x1[t]+3*x2[t]-2*x3[t],D[x3[t],t]==-x2[t]+3*x3[t]}; 
ic={x1[0]==0,x2[0]==0,x3[0]==4}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 2 e^t \left (e^{2 t}-1\right )^2 \\ \text {x2}(t)\to -2 e^t \left (e^{4 t}-1\right ) \\ \text {x3}(t)\to e^t \left (e^{2 t}+1\right )^2 \\ \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 3*x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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