9.4.22 problem problem 22

Internal problem ID [986]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 22
Date solved : Tuesday, March 04, 2025 at 12:06:52 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-5 x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{1} \left (t \right )+5 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 54

ode:=[diff(x__1(t),t) = 3*x__1(t)+2*x__2(t)+2*x__3(t), diff(x__2(t),t) = -5*x__1(t)-4*x__2(t)-2*x__3(t), diff(x__3(t),t) = 5*x__1(t)+5*x__2(t)+3*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 123

ode={D[ x1[t],t]==3*x1[t]+2*x2[t]+2*x3[t],D[ x2[t],t]==-5*x1[t]-4*x2[t]-2*x3[t],D[ x3[t],t]==5*x1[t]+5*x2[t]+3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t \left ((c_1+c_2+c_3) e^{2 t}-c_2-c_3\right ) \\ \text {x2}(t)\to e^{-2 t} \left (-\left (c_1 \left (e^{5 t}-1\right )\right )+c_2 \left (e^{3 t}-e^{5 t}+1\right )-c_3 e^{3 t} \left (e^{2 t}-1\right )\right ) \\ \text {x3}(t)\to e^{-2 t} \left (c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}-1\right )+c_3 e^{5 t}\right ) \\ \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) - 2*x__2(t) - 2*x__3(t) + Derivative(x__1(t), t),0),Eq(5*x__1(t) + 4*x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-5*x__1(t) - 5*x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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