76.25.13 problem 16

Internal problem ID [17747]
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 : 16
Date solved : Thursday, March 13, 2025 at 10:48:20 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.122 (sec). Leaf size: 72

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

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 184

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

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

✓ Sympy. Time used: 0.180 (sec). Leaf size: 58

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) - 5*x__2(t) - 3*x__3(t) + 5*x__4(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 3*x__2(t) - 2*x__3(t) + 4*x__4(t) + Derivative(x__2(t), t),0),Eq(x__2(t) + 2*x__3(t) - x__4(t) + Derivative(x__3(t), t),0),Eq(-2*x__1(t) - 4*x__2(t) - 2*x__3(t) + 5*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)
 

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