10.19.13 problem 13

Internal problem ID [1454]
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 : 13
Date solved : Saturday, March 29, 2025 at 10:55:37 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 71

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

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

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 160

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

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-\sqrt {2} t} \left (4 e^{\sqrt {2} t}+\left (\left (2+\sqrt {2}\right ) c_1+\sqrt {2} c_2\right ) e^{2 \sqrt {2} t}-\left (\left (\sqrt {2}-2\right ) c_1\right )-\sqrt {2} c_2\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-\sqrt {2} t} \left (4 e^{\sqrt {2} t}+\left (\sqrt {2} c_1-\left (\sqrt {2}-2\right ) c_2\right ) e^{2 \sqrt {2} t}-\sqrt {2} c_1+\left (2+\sqrt {2}\right ) c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.286 (sec). Leaf size: 61

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

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