\documentclass[11pt]{article}
\usepackage{epsfig}
\textheight 9.0in
\textwidth 6.0in
\hoffset -.5in
\voffset -.5in
\def\un{$\underline{\qquad\qquad\qquad}$}
\def\beqn{\begin{eqnarray*}}
\def\neqn{\end{eqnarray*}}
\def\ds{\displaystyle}
\begin{document}
\noindent 3450:436/536 Mathematical Models, Kreider, Spring 2014
\vskip .2in
\noindent Homework Set 3
\vskip .1in
\noindent Due date: Monday 24 February
\vskip .1in
Type your responses to the extent possible. If necessary, leave blank space in the document to write
equations by hand.
\vskip .1in
\begin{enumerate}
\item (20 pts)
Consider the equation $x^3 - (4+\epsilon)x+2\epsilon = 0$. Find the first order corrections to the
solutions $x_B = 0$ and $x_C=2$ with $\epsilon=.001$. Explain the odd thing that happens with $x_C$.
\item (20 pts)
For the initial value problem $y' = -y + \epsilon y^2$, $y(0)=1$, we derived the asymptotic solution
$y(\tau) = e^{-\tau} + \epsilon \left( e^{-\tau}-e^{-2\tau} \right) + \epsilon^2 \left( e^{-\tau}
-2e^{-2\tau} + e^{-3\tau} \right)$. Because this is a Bernoulli equation, it is possible to find the
exact solution, $\ds y(\tau) = { e^{-\tau} \over 1 + \epsilon (e^{-\tau}-1)}$. Use the geometric series
$\ds {1 \over 1+A} = 1 - A + A^2 + \cdots$ to show that the leading terms in the exact solution
match the asymptotic solution up to second order. This provides some measure of validation to the
perturbation approach.
\item (20 pts)
Find the first order perturbation solution to $y'(\tau) = -\epsilon y + y^2$, $y(0)=1$.
\item (20 pts)
Consider the harvesting problem $\ds {dP\over dt} = aP-bP^2-H$, $P(0)=P_0$, which we nondimensionalized to
$\ds \dot{ \tilde{P} } = \tilde{P} - \epsilon \tilde{P}^2 - \beta$, with $\epsilon = P_0/M$ and $\beta =
H/aP_0$. Suppose that the harvesting term is small, $O(\epsilon)$, so that $\beta = \alpha\epsilon$.
(a) Write both the $O(1)$ and $O(\epsilon)$ perturbation equations. (b) Use the $O(1)$ solution to
demonstrate that for very early time the population grows when $\beta$ is $O(\epsilon)$. (c) Find the
$O(\epsilon)$ solution. What feature in that solution suggests that the population might start to decline as
time goes by? (d) Using the parameter
values presented in class, $a=0.5$, $b=10^{-5}$, $P_0=1000$, determine how many fish per year, $H$, can be
harvested when $\beta = \alpha\epsilon$, assuming that $\alpha$ is 1. Note that this is a short time
limitation; as the population grows, more harvesting can safely be done.
\item 536 STUDENTS ONLY. (20 pts)
Find the first order perturbation solution to $y''(t) + 2y'(t) + \epsilon y(t) = 0$ with $y(0)=0$, $y'(0)=1$.
For the first order equation, recall that in the method of undetermined coefficients, if there is
duplication between the homogeneous solutions and the standard form of the particular solution, the
particular solution must be multiplied by the lowest power of $t$ that avoids the duplication.
\end{enumerate}
\end{document}