20.15.2 problem 2

Internal problem ID [3828]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.3, page 598
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 05:17:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -2\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 31

ode:=[diff(x__1(t),t) = -2*x__1(t)+3*x__2(t), diff(x__2(t),t) = -2*x__1(t)+5*x__2(t)]; 
ic:=x__1(0) = -2x__2(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{4 t}-3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{4 t}-{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[x1[t],t]==-2*x1[t]+3*x2[t],D[x2[t],t]==-2*x1[t]+5*x2[t]}; 
ic={x1[0]==-2,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (e^{5 t}-3\right ) \\ \text {x2}(t)\to e^{-t} \left (2 e^{5 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(2*x__1(t) - 3*x__2(t) + Derivative(x__1(t), t),0),Eq(2*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)
 

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