8.29.15 problem 16

Internal problem ID [2813]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 05:52:56 AM
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 )+1\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+5 \end{align*}

✓ Maple. Time used: 0.157 (sec). Leaf size: 60

ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t)+1, diff(x__2(t),t) = 2*x__1(t)-x__2(t)+5]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {4}{3}+{\mathrm e}^{-t} \left (\cos \left (\sqrt {2}\, t \right ) c_1 +\sin \left (\sqrt {2}\, t \right ) c_2 \right ) \\ x_{2} \left (t \right ) &= \frac {7}{3}-{\mathrm e}^{-t} \sqrt {2}\, \left (c_2 \cos \left (\sqrt {2}\, t \right )-\sin \left (\sqrt {2}\, t \right ) c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 91

ode={D[x1[t],t]==-x1[t]-x2[t],D[x2[t],t]==2*x1[t]-x2[t]+5}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to c_1 e^{-t} \cos \left (\sqrt {2} t\right )-\frac {c_2 e^{-t} \sin \left (\sqrt {2} t\right )}{\sqrt {2}}-\frac {5}{3}\\ \text {x2}(t)&\to c_2 e^{-t} \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 e^{-t} \sin \left (\sqrt {2} t\right )+\frac {5}{3} \end{align*}

✓ Sympy. Time used: 0.223 (sec). Leaf size: 128

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) - 1,0),Eq(-2*x__1(t) + x__2(t) + Derivative(x__2(t), t) - 5,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {\sqrt {2} C_{1} e^{- t} \sin {\left (\sqrt {2} t \right )}}{2} - \frac {\sqrt {2} C_{2} e^{- t} \cos {\left (\sqrt {2} t \right )}}{2} - \frac {4 \sin ^{2}{\left (\sqrt {2} t \right )}}{3} - \frac {4 \cos ^{2}{\left (\sqrt {2} t \right )}}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} \cos {\left (\sqrt {2} t \right )} - C_{2} e^{- t} \sin {\left (\sqrt {2} t \right )} + \frac {7 \sin ^{2}{\left (\sqrt {2} t \right )}}{3} + \frac {7 \cos ^{2}{\left (\sqrt {2} t \right )}}{3}\right ] \]