7.25.24 problem 24

Internal problem ID [644]
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 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 24
Date solved : Saturday, March 29, 2025 at 05:01:26 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.152 (sec). Leaf size: 86

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t)-x__3(t), diff(x__2(t),t) = -4*x__1(t)-3*x__2(t)-x__3(t), diff(x__3(t),t) = 4*x__1(t)+4*x__2(t)+2*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 103

ode={D[x1[t],t]==2*x1[t]+x2[t]-x3[t],D[x2[t],t]==-4*x1[t]-3*x2[t]-x3[t],D[x3[t],t]==4*x1[t]+4*x2[t]+2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.137 (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") 
ode=[Eq(-2*x__1(t) - x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(4*x__1(t) + 3*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - 4*x__2(t) - 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)
 

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