76.25.18 problem 22

Internal problem ID [17752]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 22
Date solved : Thursday, March 13, 2025 at 10:48:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-5 x_{1} \left (t \right )-2 x_{2} \left (t \right )-x_{3} \left (t \right )+2 x_{4} \left (t \right )+3 x_{5} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )-x_{3} \left (t \right )-x_{5} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-4 x_{4} \left (t \right )-2 x_{5} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=-3 x_{1} \left (t \right )-2 x_{2} \left (t \right )-x_{3} \left (t \right )+2 x_{4} \left (t \right )+x_{5} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.168 (sec). Leaf size: 101

ode:=[diff(x__1(t),t) = -5*x__1(t)-2*x__2(t)-x__3(t)+2*x__4(t)+3*x__5(t), diff(x__2(t),t) = -3*x__2(t), diff(x__3(t),t) = x__1(t)-x__3(t)-x__5(t), diff(x__4(t),t) = 2*x__1(t)+x__2(t)-4*x__4(t)-2*x__5(t), diff(x__5(t),t) = -3*x__1(t)-2*x__2(t)-x__3(t)+2*x__4(t)+x__5(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= c_5 \,{\mathrm e}^{-3 t} \\ x_{3} \left (t \right ) &= -c_{3} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{-2 t} \\ x_{4} \left (t \right ) &= c_5 \,{\mathrm e}^{-3 t}-c_{2} {\mathrm e}^{-4 t}-c_{1} {\mathrm e}^{-2 t} \\ x_{5} \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-2 t}+c_{1} {\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 245

ode={D[x1[t],t]==-5*x1[t]-2*x2[t]-1*x3[t]+2*x4[t]+3*x5[t],D[x2[t],t]==0*x1[t]-3*x2[t]-0*x3[t]+0*x4[t]+0*x5[t],D[x3[t],t]==1*x1[t]-0*x2[t]-1*x3[t]+0*x4[t]-x5[t],D[x4[t],t]==2*x1[t]+1*x2[t]-0*x3[t]-4*x4[t]-2*x5[t],D[x5[t],t]==-3*x1[t]-2*x2[t]-1*x3[t]+2*x4[t]+1*x5[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-4 t} \left (c_1 \left (e^{2 t}-e^{3 t}+1\right )-c_2 e^{2 t}+c_3 e^{2 t}-c_3 e^{3 t}+c_4 e^{2 t}+c_5 e^{3 t}+c_2-c_4-c_5\right ) \\ \text {x2}(t)\to c_2 e^{-3 t} \\ \text {x3}(t)\to e^{-2 t} \left (c_1 \left (e^t-1\right )+(c_3-c_5) e^t+c_5\right ) \\ \text {x4}(t)\to e^{-4 t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^t-1\right )-c_5 e^{2 t}+c_4+c_5\right ) \\ \text {x5}(t)\to e^{-4 t} \left (c_1 \left (-e^{3 t}\right )-c_2 e^{2 t}+c_3 e^{2 t}-c_3 e^{3 t}+c_4 e^{2 t}+c_5 e^{2 t}+c_5 e^{3 t}+c_1+c_2-c_4-c_5\right ) \\ \end{align*}

✓ Sympy. Time used: 0.231 (sec). Leaf size: 88

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") 
x__5 = Function("x__5") 
ode=[Eq(5*x__1(t) + 2*x__2(t) + x__3(t) - 2*x__4(t) - 3*x__5(t) + Derivative(x__1(t), t),0),Eq(3*x__2(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) + x__3(t) + x__5(t) + Derivative(x__3(t), t),0),Eq(-2*x__1(t) - x__2(t) + 4*x__4(t) + 2*x__5(t) + Derivative(x__4(t), t),0),Eq(3*x__1(t) + 2*x__2(t) + x__3(t) - 2*x__4(t) - x__5(t) + Derivative(x__5(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t),x__5(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- 4 t} + C_{4} e^{- t} + \left (C_{2} + C_{3}\right ) e^{- 2 t}, \ x^{2}{\left (t \right )} = C_{5} e^{- 3 t}, \ x^{3}{\left (t \right )} = - C_{2} e^{- 2 t} - C_{4} e^{- t}, \ x^{4}{\left (t \right )} = - C_{1} e^{- 4 t} + C_{2} e^{- 2 t} + C_{5} e^{- 3 t}, \ x^{5}{\left (t \right )} = C_{1} e^{- 4 t} + C_{3} e^{- 2 t} + C_{4} e^{- t}\right ] \]