20.20.31 problem 35

Internal problem ID [3921]
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 : 35
Date solved : Tuesday, March 04, 2025 at 05:19:29 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 59

ode={D[x1[t],t]==0*x1[t]-6*x2[t],D[x2[t],t]==1*x1[t]-5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.089 (sec). Leaf size: 34

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

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