14.24.1 problem Example 1, page 348

Internal problem ID [2748]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : Example 1, page 348
Date solved : Sunday, March 30, 2025 at 12:16:21 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.152 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 64

ode={D[ x1[t],t]==1*x1[t]+1*x2[t]+0*x3[t],D[ x2[t],t]==0*x1[t]+1*x2[t]-0*x3[t],D[ x3[t],t]==0*x1[t]-0*x2[t]+2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.086 (sec). Leaf size: 29

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) - x__2(t) + Derivative(x__1(t), t),0),Eq(-x__2(t) + Derivative(x__2(t), t),0),Eq(-2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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