14.23.2 problem 2

Internal problem ID [2741]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:16:09 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 )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.151 (sec). Leaf size: 54

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (\sin \left (t \right ) c_1 +\cos \left (t \right ) c_2 \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (2 \sin \left (t \right ) c_1 +\sin \left (t \right ) c_2 -\cos \left (t \right ) c_1 +2 \cos \left (t \right ) c_2 \right )}{5} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 120

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 \cos (t)+(2 c_1-5 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \text {x3}(t)\to c_3 e^t \\ \text {x1}(t)\to e^{-t} (c_1 \cos (t)+(2 c_1-5 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \text {x3}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__1(t) + 5*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 3*x__2(t) + Derivative(x__2(t), t),0),Eq(-x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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