14.24.6 problem 4

Internal problem ID [2753]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:16:28 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.167 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 63

ode={D[ x1[t],t]==2*x1[t]+0*x2[t]-1*x3[t]+0*x4[t],D[ x2[t],t]==0*x1[t]+2*x2[t]+1*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]-0*x2[t]+2*x3[t]-0*x4[t],D[ x4[t],t]==0*x1[t]-0*x2[t]-1*x3[t]+2*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} (c_1-c_3 t) \\ \text {x2}(t)\to e^{2 t} (c_3 t+c_2) \\ \text {x3}(t)\to c_3 e^{2 t} \\ \text {x4}(t)\to e^{2 t} (c_4-c_3 t) \\ \end{align*}

✓ Sympy. Time used: 0.174 (sec). Leaf size: 66

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) + x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-2*x__3(t) + Derivative(x__3(t), t),0),Eq(x__3(t) - 2*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)
 

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