10.19.14 problem 14

Internal problem ID [1455]
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 : 14
Date solved : Tuesday, March 04, 2025 at 12:36:13 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -2*x__1(t)+x__2(t)-2, diff(x__2(t),t) = x__1(t)-2*x__2(t)+1]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+c_1 \,{\mathrm e}^{-3 t}-1 \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-t}-c_1 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 72

ode={D[ x1[t],t]==-2*x1[t]+1*x2[t]-2,D[ x2[t],t]==1*x1[t]-2*x2[t]+1}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.149 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(2*x__1(t) - x__2(t) + Derivative(x__1(t), t) + 2,0),Eq(-x__1(t) + 2*x__2(t) + Derivative(x__2(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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