%% Study Outline for Exam 1
% First Order Ordinary Differential Equations:
% y' = f(t, y)
% A General Solution is a one-parameter family of
% functions all of which solve the equation.
% If given initial data, i.e, an IVP
% y' = f(t, y), y(t_0) = y_0,
% a solution is a function that satisfies both the
% differential equation and the initial condition.
%% 1. Three formula techniques:
% Linear
% Separable
% Exact.
% You should be able to recognize the three types,
% find the general solution for each, determine
% the constant of integration if initial data is given,
% and solve for the solution function explicitly as a
% function of t whenever possible. In that case,
% be able to determine the interval of definition
% of the solution function.
%% 2. Graphical techniques:
% Draw a rough direction field by hand -- using the
% technique of drawing slope segments along the
% level curves of f. Sketch in solution curves.
% For autonomous equations, that is, y' = f(y), be able
% to sketch the solution curves purely from the nature
% of the graph of f(y), and even more boldly, just from the
% zeros of f and the sign of f in between its zeros. Draw
% the Phase Line portrait on the y-axis and describe the
% stabiility properties of the stationary solutions.
%% 3. Existence and Uniqueness:
% Understand what the theorem says and its
% implication for overlap of solution curves.
%% 4. Models
% Work a problem based on one of the four first
% order models that we studied, namely:
% Continuous compound interest accounts
% Mixing problems
% Falling body
% Population Dynamics
%% 5. Stability:
% Understand the difference between stability,
% asymptotic stability and instability. Use
% Theorem 5.2 to determine for a given equation,
% the regions where stability applies. Apply
% especially in the case of autonomous equations,
% y' = f(y).
% In particular, (as mentioned above) be able to
% draw the solution curves for an autonomous
% equation solely from an understanding of the
% graph of the curve f(y).
%% 6. Numerical Technique
% Be able to apply the Euler Method
% y_(n+1) = y_n + h(f(t_n, y_n))
% to get a numerical approximation to the solution
% of the IVP
% y' = f(t, y), y(t_0) = y_0
% at some point to the right of t_0.
%% 7. Qualitative Technique
% Be able to apply the technique whereby if given
% an IVP, say y' = f(t, y), y(0) = 1, then if on
% the interval [0,1], you have
% g(t,y) <= f(t, y) <= h(t,y)
% then the solution to the original IVP must lie
% between the solutions to the IVPs
% y' = g(t,y), y(0) = 1 and y' = h(t,y), y(0) = 1.
% Presuming you can solve the latter two IVPs,
% you can then say something about the solution
% to the original on the interval [0,1].
%% Matlab
% You will not be asked to write any Matlab code,
% but you may be expected to interpret a Matlab
% session.