20.20.15 problem 15

Internal problem ID [3905]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 15
Date solved : Tuesday, March 04, 2025 at 05:19:10 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-17 x_{1} \left (t \right )-42 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-7 x_{1} \left (t \right )+4 x_{2} \left (t \right )-14 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=7 x_{1} \left (t \right )+18 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = -17*x__1(t)-42*x__3(t), diff(x__2(t),t) = -7*x__1(t)+4*x__2(t)-14*x__3(t), diff(x__3(t),t) = 7*x__1(t)+18*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+c_3 \,{\mathrm e}^{-3 t} \\ x_{2} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{4 t}}{3}+\frac {c_3 \,{\mathrm e}^{-3 t}}{3}+c_{1} {\mathrm e}^{4 t} \\ x_{3} \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{4 t}}{2}-\frac {c_3 \,{\mathrm e}^{-3 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 105

ode={D[x1[t],t]==-17*x1[t]-0*x2[t]-42*x3[t],D[x2[t],t]==-7*x1[t]+4*x2[t]-14*x3[t],D[x3[t],t]==7*x1[t]+0*x2[t]+18*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} \left (c_1 \left (3-2 e^{7 t}\right )-6 c_3 \left (e^{7 t}-1\right )\right ) \\ \text {x2}(t)\to e^{-3 t} \left (-\left (c_1 \left (e^{7 t}-1\right )\right )+(c_2-2 c_3) e^{7 t}+2 c_3\right ) \\ \text {x3}(t)\to e^{-3 t} \left (c_1 \left (e^{7 t}-1\right )+c_3 \left (3 e^{7 t}-2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(17*x__1(t) + 42*x__3(t) + Derivative(x__1(t), t),0),Eq(7*x__1(t) - 4*x__2(t) + 14*x__3(t) + Derivative(x__2(t), t),0),Eq(-7*x__1(t) - 18*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - 3 C_{1} e^{- 3 t} - 2 C_{2} e^{4 t}, \ x^{2}{\left (t \right )} = - C_{1} e^{- 3 t} + C_{3} e^{4 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{4 t}\right ] \]