7.24.17 problem 27 and 36

Internal problem ID [617]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.3 (Matrices and linear systems). Problems at page 364
Problem number : 27 and 36
Date solved : Saturday, March 29, 2025 at 05:00:49 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 10\\ x_{2} \left (0\right ) = 12\\ x_{3} \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.188 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = x__2(t)+x__3(t), diff(x__2(t),t) = x__1(t)+x__3(t), diff(x__3(t),t) = x__1(t)+x__2(t)]; 
ic:=x__1(0) = 10x__2(0) = 12x__3(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 7 \,{\mathrm e}^{2 t}+3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= 7 \,{\mathrm e}^{2 t}+5 \,{\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= 7 \,{\mathrm e}^{2 t}-8 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 205

ode={D[x1[t],t]==x2[t]+3*x3[t],D[x2[t],t]==x1[t]+x3[t],D[x3[t],t]==x1[t]+x2[t]}; 
ic={x1[0]==10,x2[0]==12,x3[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{34} e^{-\frac {1}{2} \left (2+\sqrt {17}\right ) t} \left (\left (391+31 \sqrt {17}\right ) e^{3 t/2}-442 e^{\frac {\sqrt {17} t}{2}}+\left (391-31 \sqrt {17}\right ) e^{\left (\frac {3}{2}+\sqrt {17}\right ) t}\right ) \\ \text {x2}(t)\to -\frac {1}{34} e^{-\frac {1}{2} \left (2+\sqrt {17}\right ) t} \left (\left (17+45 \sqrt {17}\right ) e^{3 t/2}-442 e^{\frac {\sqrt {17} t}{2}}+\left (17-45 \sqrt {17}\right ) e^{\left (\frac {3}{2}+\sqrt {17}\right ) t}\right ) \\ \text {x3}(t)\to \frac {1}{34} e^{\frac {1}{2} \left (t-\sqrt {17} t\right )} \left (\left (45 \sqrt {17}-17\right ) e^{\sqrt {17} t}-17-45 \sqrt {17}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 42

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

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