10.19.11 problem 11

Internal problem ID [1452]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 9.1, The Phase Plane: Linear Systems. page 505
Problem number : 11
Date solved : Saturday, March 29, 2025 at 10:55:34 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.105 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 65

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

\begin{align*} \text {x1}(t)\to c_1 e^{-t} \\ \text {x2}(t)\to c_2 e^{-t} \\ \text {x1}(t)\to c_1 e^{-t} \\ \text {x2}(t)\to 0 \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to c_2 e^{-t} \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.060 (sec). Leaf size: 14

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

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