7.24.20 problem 30 and 39

Internal problem ID [620]
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 : 30 and 39
Date solved : Saturday, March 29, 2025 at 05:00:53 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )-12 x_{2} \left (t \right )-x_{3} \left (t \right )-6 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-4 x_{2} \left (t \right )-x_{4} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.187 (sec). Leaf size: 48

ode:=[diff(x__1(t),t) = x__1(t)-4*x__2(t)-2*x__4(t), diff(x__2(t),t) = x__2(t), diff(x__3(t),t) = 6*x__1(t)-12*x__2(t)-x__3(t)-6*x__4(t), diff(x__4(t),t) = -4*x__2(t)-x__4(t)]; 
ic:=x__1(0) = 1x__2(0) = 1x__3(0) = 1x__4(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= -2 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \\ x_{3} \left (t \right ) &= -6 \,{\mathrm e}^{t}+7 \,{\mathrm e}^{-t} \\ x_{4} \left (t \right ) &= -2 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 56

ode={D[x1[t],t]==x1[t]-4*x2[t]-2*x4[t],D[x2[t],t]==x2[t],D[x3[t],t]==6*x1[t]-12*x2[t]-x3[t]-6*x4[t],D[x4[t],t]==-4*x2[t]-x4[t]}; 
ic={x1[0]==1,x2[0]==1,x3[0]==1,x4[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 3 e^{-t}-2 e^t \\ \text {x2}(t)\to e^t \\ \text {x3}(t)\to 7 e^{-t}-6 e^t \\ \text {x4}(t)\to 3 e^{-t}-2 e^t \\ \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(-x__1(t) + 4*x__2(t) + 2*x__4(t) + Derivative(x__1(t), t),0),Eq(-x__2(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) + 12*x__2(t) + x__3(t) + 6*x__4(t) + Derivative(x__3(t), t),0),Eq(4*x__2(t) + x__4(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- t} + \frac {C_{2} e^{t}}{3}, \ x^{2}{\left (t \right )} = - \frac {C_{3} e^{t}}{2}, \ x^{3}{\left (t \right )} = C_{2} e^{t} + C_{4} e^{- t}, \ x^{4}{\left (t \right )} = C_{1} e^{- t} + C_{3} e^{t}\right ] \]