%% 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.