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\flushleft \underline{MATH 3504 BONUS problem}\\
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1. Consider the nonlinear initial value problem
\[ \left\{ \begin{array}{ll}
x' = \underbrace{x(y-1)}_{=f(t,x,y)} \\
y' = \underbrace{4-x^2-y^2}_{=g(t,x,y)} \\
x(0)=x_0, \quad y(0)=y_0.
\end{array} \right.\]
(note: \textit{the $f$ and $g$ functions do not depend on $t$ (this is typical of the types of nonlinear problems that we solved in class)}) \\
The Euler method for this system works as it does in the one-dimensional case; we just need to do it twice: for some stepsize $h$,
\[ \left\{ \begin{array}{ll}
x(t_{n+1})=x(t_n)+h f(t_n,x(t_n),y(t_n)) \\
y(t_{n+1})=y(t_n)+h g(t_n,x(t_n),y(t_n))
\end{array} \right.\]
Let the initial condition be defined by your student $F$-number where
\[ \left\{ \begin{array}{ll}
x_0=-\dfrac{\text{2nd to last digit of your F-number}}{5} \\
y_0=\dfrac{\text{last digit of your F-number}}{5}.
\end{array} \right.\]
(for example: if your F number is F00000012, then take $x_0=-\dfrac{1}{5}$ and $y_0=\dfrac{2}{5}$).
\begin{enumerate}[1.]
\item Use Euler's method with $h=0.01$ to solve the system for $t \in [0,20]$. Submit the spreadsheet you used.
\item Plot the solutions $y(t)$ and $x(t)$ as functions of $t$ (i.e. the horizontal axis is the $t$ variable and the vertical axis are the solutions). Submit the picture you get.
\item Plot the phase diagram of this solution (i.e. the horizontal axis is the $x$ variable and the vertical axis is the $y$ variable). Submit the picture you get.
\item What kind of stability appears to be occurring?
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