20.13.11 problem 11

Internal problem ID [3820]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 05:17:32 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 )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 3 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 27

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t), diff(x__2(t),t) = -x__1(t)+4*x__2(t)]; 
ic:=x__1(0) = 1x__2(0) = 3; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 30

ode={D[x1[t],t]==2*x1[t]+x2[t],D[x2[t],t]==-x1[t]+4*x2[t]}; 
ic={x1[0]==1,x2[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} (2 t+1) \\ \text {x2}(t)\to e^{3 t} (2 t+3) \\ \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 39

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),0),Eq(x__1(t) - 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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