20.20.28 problem 28

Internal problem ID [3918]
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 : 28
Date solved : Tuesday, March 04, 2025 at 05:19:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right )+3 x_{3} \left (t \right )+{\mathrm e}^{6 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=-9 x_{1} \left (t \right )-3 x_{2} \left (t \right )-9 x_{3} \left (t \right )+1\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )+4 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.088 (sec). Leaf size: 93

ode:=[diff(x__1(t),t) = 2*x__1(t)-4*x__2(t)+3*x__3(t)+exp(6*t), diff(x__2(t),t) = -9*x__1(t)-3*x__2(t)-9*x__3(t)+1, diff(x__3(t),t) = 4*x__1(t)+4*x__2(t)+3*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {4 \,{\mathrm e}^{6 t}}{63}-\frac {4}{3}-{\mathrm e}^{-t} c_{1} -c_{2} {\mathrm e}^{6 t}+t \,{\mathrm e}^{6 t}+2 c_3 \,{\mathrm e}^{-3 t} \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{6 t}-t \,{\mathrm e}^{6 t}+\frac {{\mathrm e}^{6 t}}{9}+\frac {1}{3} \\ x_{3} \left (t \right ) &= \frac {4 \,{\mathrm e}^{6 t}}{63}+\frac {4}{3}+{\mathrm e}^{-t} c_{1} -2 c_3 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 149

ode={D[x1[t],t]==2*x1[t]-4*x2[t]+3*x3[t]+Exp[6*t],D[x2[t],t]==-9*x1[t]-3*x2[t]-9*x3[t]+1,D[x3[t],t]==4*x1[t]+4*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^{6 t} \left (t-\frac {4}{63}+c_1+c_3\right )+2 (c_1+c_2+c_3) e^{-3 t}-(2 c_1+2 c_2+3 c_3) e^{-t}-\frac {4}{3} \\ \text {x2}(t)\to (c_1+c_2+c_3) e^{-3 t}-\frac {1}{9} e^{6 t} (9 t-1+9 c_1+9 c_3)+\frac {1}{3} \\ \text {x3}(t)\to \frac {4 e^{6 t}}{63}-2 (c_1+c_2+c_3) e^{-3 t}+(2 c_1+2 c_2+3 c_3) e^{-t}+\frac {4}{3} \\ \end{align*}

✓ Sympy. Time used: 0.291 (sec). Leaf size: 90

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) + 4*x__2(t) - 3*x__3(t) - exp(6*t) + Derivative(x__1(t), t),0),Eq(9*x__1(t) + 3*x__2(t) + 9*x__3(t) + Derivative(x__2(t), t) - 1,0),Eq(-4*x__1(t) - 4*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)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{- t} - C_{2} e^{- 3 t} + t e^{6 t} - \left (C_{3} + \frac {4}{63}\right ) e^{6 t} - \frac {4}{3}, \ x^{2}{\left (t \right )} = - \frac {C_{2} e^{- 3 t}}{2} - t e^{6 t} + \left (C_{3} + \frac {1}{9}\right ) e^{6 t} + \frac {1}{3}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{- 3 t} + \frac {4 e^{6 t}}{63} + \frac {4}{3}\right ] \]