70.25.17 problem 21

Internal problem ID [19116]
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 : 21
Date solved : Thursday, October 02, 2025 at 03:38:08 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+2 x_{2} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+3 x_{2} \left (t \right )-x_{3} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-2 x_{2} \left (t \right )-4 x_{3} \left (t \right )+2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-7 x_{1} \left (t \right )+x_{2} \left (t \right )-7 x_{3} \left (t \right )+3 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.194 (sec). Leaf size: 80

ode:=[diff(x__1(t),t) = -2*x__1(t)+2*x__2(t)-2*x__4(t), diff(x__2(t),t) = -x__1(t)+3*x__2(t)-x__3(t)+x__4(t), diff(x__3(t),t) = -2*x__1(t)-2*x__2(t)-4*x__3(t)+2*x__4(t), diff(x__4(t),t) = -7*x__1(t)+x__2(t)-7*x__3(t)+3*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 241

ode={D[x1[t],t]==-2*x1[t]+2*x2[t]+0*x3[t]-2*x4[t],D[x2[t],t]==-1*x1[t]+3*x2[t]-1*x3[t]+1*x4[t],D[x3[t],t]==-2*x1[t]-2*x2[t]-4*x3[t]+2*x4[t],D[x4[t],t]==-7*x1[t]+1*x2[t]-7*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}{2} e^{-4 t} \left (c_1 \left (-e^{2 t}+e^{6 t}+2\right )+(c_2+c_3-c_4) e^{6 t}+(-c_2-3 c_3+c_4) e^{2 t}+2 c_3\right )\\ \text {x2}(t)&\to -\frac {1}{2} e^{2 t} \left (c_1 \left (e^{2 t}-1\right )-c_2 \left (e^{2 t}+1\right )+(c_3-c_4) \left (e^{2 t}-1\right )\right )\\ \text {x3}(t)&\to \frac {1}{2} e^{-2 t} \left (c_1 \left (-e^{4 t}\right )-c_2 e^{4 t}-c_3 e^{4 t}+c_4 e^{4 t}+c_1+c_2+3 c_3-c_4\right )\\ \text {x4}(t)&\to \frac {1}{2} e^{-4 t} \left (-\left (c_1 \left (e^{6 t}+e^{8 t}-2\right )\right )-(c_2+c_3-c_4) e^{6 t}+(c_2-c_3+c_4) e^{8 t}+2 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 75

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(2*x__1(t) - 2*x__2(t) + 2*x__4(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 3*x__2(t) + x__3(t) - x__4(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) + 2*x__2(t) + 4*x__3(t) - 2*x__4(t) + Derivative(x__3(t), t),0),Eq(7*x__1(t) - x__2(t) + 7*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)
 

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