Solve the system of differential equations using the Laplace Transform. $\{$dy$1=-0\mathrm{.}3186089264*$y$2+0\mathrm{.}007133565939*$y$1-1\mathrm{.}232409966$; dy$2=-0\mathrm{.}3491081494*$y$1-0\mathrm{.}9708343724*$y$2-1\mathrm{.}020199009\}$

If relevant, I offer you the following information. Python code used to solve the system of equations using the Runge$-$Kutta method of order $4$:

import numpy as np

import matplotlib.pyplot as plt

# Definition of the ODE system

def ode$\_$system$($t$,$ y$)$:

y$1,$ y$2=$ y

dy$1=-0\mathrm{.}3186089264*$y$2+0\mathrm{.}007133565939*$y$1-1\mathrm{.}232409966$

dy$2=-0\mathrm{.}3491081494*$y$1-0\mathrm{.}9708343724*$y$2-1\mathrm{.}020199009$

return $[$dy$1,$ dy$2]$

# Implementation of the Runge$-$Kutta method of order $4$

def runge$\_$kutta$\_4($func$,$ y$0,$ t$0,$ tf$,$ h$)$:

t$\_$values $=$ np$.$arange$($t$0,$ tf $+$ h$,$ h$)$ n $=$ len$($t$\_$values$)$ y$\_$values $=$ np$.$zeros$(($n$,$ len$($y$0)))$ y$\_$values$[0]=$ y$0$ for i in range$(1,$ n$)$: k$1=$ h $*$ np$.$array$($func$($t$\_$values$[$i $-1],$ y$\_$values$[$i $-1]))$ k$2=$ h $*$ np$.$array$($func$($ t$\_$values$[$i $-1]+$ h $/2,$ y$\_$values$[$i $-1]+$ k$1/2))$ k$3=$ h $*$ np$.$array$($func$($t$\_$values$[$i $-1]+$ h $/2,$ y$\_$values$[$i $-1]+$ k$2/2))$ k$4=$ h $*$ np$.$array$($func$($t$\_$values$[$i $-1]+$ h$,$ y$\_$values$[$i $-1]+$ k$3))$

y$\_$values$[$i$]=$ y$\_$values$[$i $-1]+($k$1+2*$ k$2+2*$ k$3+$ k$4)/6$

return t$\_$values, y$\_$values

# Initial conditions and parameters

y$0=[3\mathrm{.}75627739,1\mathrm{.}98468425]$

t$0=0$

tf $=20$

h $=0\mathrm{.}1$

# Solving the system of ODEs

t$,$ solution $=$ runge$\_$kutta$\_4($system$\_$ode, y$0,$ t$0,$ tf$,$ h$)$

# Displaying the results

plt$.$plot$($t$,$ solution$[$:$,0],$ label$=\text{'}$y$1($t$)\text{'})$

plt$.$plot$($t$,$ solution$[$:$,1],$ label$=\text{'}$y$2($t$)\text{'})$

plt$.$xlabel$(\text{'}$Time $($t$)\text{'})$

plt$.$ylabel$(\text{'}$Solutions$\text{'})$

plt$.$legend$()$

plt$.$title$(\text{'}$Solution of the Runge$-$Kutta ODE System of Order $4\text{'})$

plt$.$show$()$

Resulting graph $($image$)$ Soluci$$n del Sistema de EDOs con Runge$-$Kutta de Orden $4$