%% 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.