%% Study Outline for Exam 2
% Second Order Linear Differential Equations:
% y'' + p(t)y’ + q(t)y = g(t) [*] (Inhomogeneous)
% y'' + p(t)y’ + q(t)y = 0 [**] (Homogeneous)
% where p, q and g are continuous functions on an interval.
%% 1. Existence and Uniqueness of solutions to IVP:
% y(t_0) = y_0, y’(t_0) = y’_0. [***]
% For any t_0 in the interval on which the coeff fctns are
% continuous, there is exactly one solution of [*]
% satisfying the initial condtions [***]. The same is
% true with [**].
%% 2. The set of solutions to [**] is a
% two-dimensional vector space,
% meaning that there are two linearly independent (neither
% is a multiple of the other) solutions so that EVERY
% solution is a linear combination of those two.
% Two solutions y_1 and y_2 are linearly independent and
% so form a basis for the solution set exactly when the
% Wronskian, which is given by the following formula and
% enjoys the property that it is either identically zero
% or never zero:
% W = y_1 y’_2 – y’_1 y_2
% is NOT zero.
%% 3. Constant Coefficient Homogeneous Equations
% ay’’ + by’ + cy = 0.
% (a) The characteristic polynomial is
% ar^2 + br + c;
% Its roots determine the solutions.
% Distinct Real Roots: r_1, r_2 give
% e^(r_1 t), e^(r_2 t)
% Complex Conjugate Roots: alpha +_ i beta give
% e^(alpha t)*cos(beta t), e^(alpha t)*sin(beta t)
% Repeated Real Root: r_0 gives
% e^(r_0 t), te^(r_0 t).
%% 4. Reduction of Order
% If y_1(t) is a solution of [**], then get a second
% linearly independent solution by substituting
% y_2(t) = u(t)y_1(t)
% into [**], noting that the result does not depend on u,
% then solving the resulting differential equation for u'
% and then integrating to get u.
%% 5. Inhomogeneous Equations
% (a) Undetermined Coefficients (only for constant coeff
% eqns). See table on p. 181. Don't forget to multiply
% 'candidate' by t^s, where s is the smallest integer
% guaranteeing that no term in the candidate is a solution
% of the homogeneous eqn.
% (b) Variation of Parameters (requires eqn to be
% normalized).
% If y_1 and y_2 are lin ind sols of homog eqn, then
% -y_1(t) int^t y_2(s)g(s)/W(s) ds +
% y_2(t) int^t y_1(s)g(s)/W(s) ds
% is a sol of the inhomog eqn.
%% 6. Mass-Spring System
% Damped or Undamped
% i.e., mu'' + ku = 0 or mu'' + gamma u' + ku = 0.
% Resonance and Beats
% i.e., mu'' + ku = Rcos(omega t), omega_0 = sqrt(k/m).
% Resonance for omega = omega_0;
% Beats for omega very close to omega_0.