20.20.25 problem 25

Internal problem ID [3915]
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 : 25
Date solved : Tuesday, March 04, 2025 at 05:19:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-6 x_{1} \left (t \right )+x_{2} \left (t \right )+1\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{1} \left (t \right )-5 x_{2} \left (t \right )+{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = -6*x__1(t)+x__2(t)+1, diff(x__2(t),t) = 6*x__1(t)-5*x__2(t)+exp(-t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-8 t} c_{2} +c_{1} {\mathrm e}^{-3 t}+\frac {{\mathrm e}^{-t}}{14}+\frac {5}{24} \\ x_{2} \left (t \right ) &= -2 \,{\mathrm e}^{-8 t} c_{2} +3 c_{1} {\mathrm e}^{-3 t}+\frac {5 \,{\mathrm e}^{-t}}{14}+\frac {1}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 100

ode={D[x1[t],t]==-6*x1[t]+1*x2[t]+1,D[x2[t],t]==6*x1[t]-5*x2[t]+Exp[-t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {e^{-t}}{14}+\frac {1}{5} (3 c_1-c_2) e^{-8 t}+\frac {1}{5} (2 c_1+c_2) e^{-3 t}+\frac {5}{24} \\ \text {x2}(t)\to \frac {5 e^{-t}}{14}+\frac {2}{5} (c_2-3 c_1) e^{-8 t}+\frac {3}{5} (2 c_1+c_2) e^{-3 t}+\frac {1}{4} \\ \end{align*}

✓ Sympy. Time used: 0.239 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(6*x__1(t) - x__2(t) + Derivative(x__1(t), t) - 1,0),Eq(-6*x__1(t) + 5*x__2(t) + Derivative(x__2(t), t) - exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- 3 t}}{3} - \frac {C_{2} e^{- 8 t}}{2} + \frac {5}{24} + \frac {e^{- t}}{14}, \ x^{2}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{- 8 t} + \frac {1}{4} + \frac {5 e^{- t}}{14}\right ] \]