70.25.16 problem 20

Internal problem ID [19115]
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 : 20
Date solved : Thursday, October 02, 2025 at 03:38:07 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+8 x_{2} \left (t \right )+5 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+16 x_{2} \left (t \right )+10 x_{3} \left (t \right )+6 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{1} \left (t \right )-14 x_{2} \left (t \right )-11 x_{3} \left (t \right )-3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-x_{1} \left (t \right )-8 x_{2} \left (t \right )-5 x_{3} \left (t \right )-3 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.147 (sec). Leaf size: 88

ode:=[diff(x__1(t),t) = x__1(t)+8*x__2(t)+5*x__3(t)+3*x__4(t), diff(x__2(t),t) = 2*x__1(t)+16*x__2(t)+10*x__3(t)+6*x__4(t), diff(x__3(t),t) = 5*x__1(t)-14*x__2(t)-11*x__3(t)-3*x__4(t), diff(x__4(t),t) = -x__1(t)-8*x__2(t)-5*x__3(t)-3*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{9 t}+c_4 \,{\mathrm e}^{-6 t} \\ x_{2} \left (t \right ) &= 2 c_2 +2 c_3 \,{\mathrm e}^{9 t}+2 c_4 \,{\mathrm e}^{-6 t}+c_1 \\ x_{3} \left (t \right ) &= -c_3 \,{\mathrm e}^{9 t}-4 c_4 \,{\mathrm e}^{-6 t}-c_2 -c_1 \\ x_{4} \left (t \right ) &= -c_3 \,{\mathrm e}^{9 t}-c_4 \,{\mathrm e}^{-6 t}-4 c_2 -c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 311

ode={D[x1[t],t]==1*x1[t]+8*x2[t]+5*x3[t]+3*x4[t],D[x2[t],t]==2*x1[t]+16*x2[t]+10*x3[t]+6*x4[t],D[x3[t],t]==5*x1[t]-14*x2[t]-11*x3[t]-3*x4[t],D[x4[t],t]==-1*x1[t]-8*x2[t]-5*x3[t]-3*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{3} e^{-6 t} \left (c_1 \left (e^{6 t}+e^{15 t}+1\right )+c_2 \left (-e^{6 t}+2 e^{15 t}-1\right )+c_3 e^{15 t}-c_4 e^{6 t}+c_4 e^{15 t}-c_3\right )\\ \text {x2}(t)&\to \frac {1}{3} e^{-6 t} \left (2 c_1 \left (-2 e^{6 t}+e^{15 t}+1\right )+c_2 \left (e^{6 t}+4 e^{15 t}-2\right )+2 c_3 e^{15 t}-2 c_4 e^{6 t}+2 c_4 e^{15 t}-2 c_3\right )\\ \text {x3}(t)&\to \frac {1}{3} e^{-6 t} \left (-\left (c_1 \left (-5 e^{6 t}+e^{15 t}+4\right )\right )-2 c_2 \left (e^{6 t}+e^{15 t}-2\right )-c_3 e^{15 t}+c_4 e^{6 t}-c_4 e^{15 t}+4 c_3\right )\\ \text {x4}(t)&\to \frac {1}{3} e^{-6 t} \left (-\left (c_1 \left (-2 e^{6 t}+e^{15 t}+1\right )\right )+c_2 \left (e^{6 t}-2 e^{15 t}+1\right )-c_3 e^{15 t}+4 c_4 e^{6 t}-c_4 e^{15 t}+c_3\right ) \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 87

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) - 8*x__2(t) - 5*x__3(t) - 3*x__4(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 16*x__2(t) - 10*x__3(t) - 6*x__4(t) + Derivative(x__2(t), t),0),Eq(-5*x__1(t) + 14*x__2(t) + 11*x__3(t) + 3*x__4(t) + Derivative(x__3(t), t),0),Eq(x__1(t) + 8*x__2(t) + 5*x__3(t) + 3*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)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1}}{3} + \frac {C_{2}}{3} - C_{3} e^{- 6 t} - C_{4} e^{9 t}, \ x^{2}{\left (t \right )} = - \frac {C_{1}}{3} - \frac {2 C_{2}}{3} - 2 C_{3} e^{- 6 t} - 2 C_{4} e^{9 t}, \ x^{3}{\left (t \right )} = C_{2} + 4 C_{3} e^{- 6 t} + C_{4} e^{9 t}, \ x^{4}{\left (t \right )} = C_{1} + C_{3} e^{- 6 t} + C_{4} e^{9 t}\right ] \]