70.26.14 problem 18

Internal problem ID [19132]
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 6.4 (Nondefective Matrices with Complex Eigenvalues). Problems at page 419
Problem number : 18
Date solved : Thursday, October 02, 2025 at 03:38:31 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.157 (sec). Leaf size: 130

ode:=[diff(x__1(t),t) = 3*x__2(t)-2*x__4(t), diff(x__2(t),t) = -1/2*x__1(t)+x__2(t)-3*x__3(t)-5/2*x__4(t), diff(x__3(t),t) = 3*x__2(t)-5*x__3(t)-3*x__4(t), diff(x__4(t),t) = x__1(t)+3*x__2(t)-3*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 261

ode={D[x1[t],t]==0*x1[t]+3*x2[t]+0*x3[t]-2*x4[t],D[x2[t],t]==-1/2*x1[t]+1*x2[t]-3*x3[t]-5/2*x4[t],D[x3[t],t]==0*x1[t]+3*x2[t]-5*x3[t]-3*x4[t],D[x4[t],t]==1*x1[t]+3*x2[t]+0*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^{-2 t} \left (c_1 e^t-c_4 e^t+2 (-c_2+c_3+c_4) \cos (3 t)+(c_1+2 c_2-c_4) \sin (3 t)+c_1+2 c_2-2 c_3-c_4\right )\\ \text {x2}(t)&\to \frac {1}{2} e^{-2 t} \left ((c_4-c_1) e^t+(c_1+2 c_2-c_4) \cos (3 t)+2 (c_2-c_3-c_4) \sin (3 t)\right )\\ \text {x3}(t)&\to \frac {1}{2} e^{-2 t} ((c_1+2 c_2-c_4) \cos (3 t)+2 (c_2-c_3-c_4) \sin (3 t)-c_1-2 c_2+2 c_3+c_4)\\ \text {x4}(t)&\to \frac {1}{2} e^{-2 t} \left (c_1 \left (-e^t\right )+c_4 e^t+2 (-c_2+c_3+c_4) \cos (3 t)+(c_1+2 c_2-c_4) \sin (3 t)+c_1+2 c_2-2 c_3-c_4\right ) \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 139

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

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