7.22.11 problem 26

Internal problem ID [586]
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 5.1 (First order systems and applications). Problems at page 335
Problem number : 26
Date solved : Saturday, March 29, 2025 at 04:57:19 PM
CAS classification : system_of_ODEs

\begin{align*} 10 \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+x_{3} \left (t \right )\\ 10 \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ 10 \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.155 (sec). Leaf size: 166

ode:=[10*diff(x__1(t),t) = -x__1(t)+x__3(t), 10*diff(x__2(t),t) = x__1(t)-x__2(t), 10*diff(x__3(t),t) = x__2(t)-x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-\frac {3 t}{20}} \sin \left (\frac {\sqrt {3}\, t}{20}\right )+c_3 \,{\mathrm e}^{-\frac {3 t}{20}} \cos \left (\frac {\sqrt {3}\, t}{20}\right ) \\ x_{2} \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-\frac {3 t}{20}} \sin \left (\frac {\sqrt {3}\, t}{20}\right )}{2}-\frac {c_2 \,{\mathrm e}^{-\frac {3 t}{20}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{20}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {3 t}{20}} \cos \left (\frac {\sqrt {3}\, t}{20}\right )}{2}+\frac {c_3 \,{\mathrm e}^{-\frac {3 t}{20}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{20}\right )}{2}+c_1 \\ x_{3} \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-\frac {3 t}{20}} \sin \left (\frac {\sqrt {3}\, t}{20}\right )}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {3 t}{20}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{20}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {3 t}{20}} \cos \left (\frac {\sqrt {3}\, t}{20}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {3 t}{20}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{20}\right )}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 235

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-3 t/20} \left ((c_1+c_2+c_3) e^{3 t/20}+(2 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} t}{20}\right )-\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} t}{20}\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-3 t/20} \left ((c_1+c_2+c_3) e^{3 t/20}-(c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} t}{20}\right )+\sqrt {3} (c_1-c_3) \sin \left (\frac {\sqrt {3} t}{20}\right )\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-3 t/20} \left ((c_1+c_2+c_3) e^{3 t/20}-(c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} t}{20}\right )-\sqrt {3} (c_1-c_2) \sin \left (\frac {\sqrt {3} t}{20}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.261 (sec). Leaf size: 163

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__3(t) + 10*Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) + 10*Derivative(x__2(t), t),0),Eq(-x__2(t) + x__3(t) + 10*Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} - \left (\frac {C_{2}}{2} - \frac {\sqrt {3} C_{3}}{2}\right ) e^{- \frac {3 t}{20}} \cos {\left (\frac {\sqrt {3} t}{20} \right )} + \left (\frac {\sqrt {3} C_{2}}{2} + \frac {C_{3}}{2}\right ) e^{- \frac {3 t}{20}} \sin {\left (\frac {\sqrt {3} t}{20} \right )}, \ x^{2}{\left (t \right )} = C_{1} - \left (\frac {C_{2}}{2} + \frac {\sqrt {3} C_{3}}{2}\right ) e^{- \frac {3 t}{20}} \cos {\left (\frac {\sqrt {3} t}{20} \right )} - \left (\frac {\sqrt {3} C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{- \frac {3 t}{20}} \sin {\left (\frac {\sqrt {3} t}{20} \right )}, \ x^{3}{\left (t \right )} = C_{1} + C_{2} e^{- \frac {3 t}{20}} \cos {\left (\frac {\sqrt {3} t}{20} \right )} - C_{3} e^{- \frac {3 t}{20}} \sin {\left (\frac {\sqrt {3} t}{20} \right )}\right ] \]