Estimating the Position of a Moving Dog Using Euler's Method
Given the initial condition \( x_0 = 0, y_0 = 0 \), and \( D = 0.5 \), we want to estimate \( x_n, y_n \) using Euler's method with a step length of \( h = 0.5 \). Euler's method gives us \( x(n+1) = x(n) + h \cdot f(x(n), y(n)) \) and similarly for \( y(n+1) \).
(i)
For \( n = 0 \) (at \( t = 0 \)):
<p>\( x(1) = x(0) + h \cdot f(x(0), y(0)) \)</p>
<p>\( x(1) = 0 + 0.5 \cdot (0.5 \cdot \cos(1-0)) \)</p>
<p>\( x(1) = 0.25 \cdot \cos(1) \)</p>
<p>\( y(1) = y(0) + h \cdot g(x(0), y(0)) \)</p>
<p>\( y(1) = 0 + 0.5 \cdot (0.5 \cdot \sin(1-0)) \)</p>
<p>\( y(1) = 0.25 \cdot \sin(1) \)</p>
For \( n = 1 \) (at \( t = 0.5 \)):
<p>\( x(2) = x(1) + h \cdot f(x(1), y(1)) \)</p>
<p>\( y(2) = y(1) + h \cdot g(x(1), y(1)) \)</p>
And so on, for each successive \( n \).
Calculate \( x(n) \) and \( y(n) \) for \( n = 36, 37, 38 \) to get the position at \( t = 18, 19, 20 \) seconds respectively.
(ii)
To find the distance from the center of the pond:
<p>\( d = \sqrt{x(n)^2 + y(n)^2} \)</p>
Use the values of \( x(n) \) and \( y(n) \) computed in part (i).
(iii)
Analyze the direction and magnitude of \( x(n) \) and \( y(n) \) changes as \( n \to \infty \) to predict the long-term behavior.
Please note that without a calculator or more detailed calculations, exact numerical values cannot be provided, and the number of steps required for the Euler's method here is excessively large for manual calculation. This is a numerical approximation method typically executed with the help of a computer or a calculator with programming features.
