FINAL EXAM TOPICS
Exam is comprehensive. It will focus on topics marked in bold.
- First-order equations: integrating factors, separation of variables, variation of parameters, existence and uniqueness theorems (linear and non-linear equations), exact equations and integrating factors, change of variables: Bernoulli equations.
- Autonomous equations: logistic equations, equilibrium points and stability.
- Second-order linear equations: characteristic polynomial (distinct roots only), principle of superposition: fundamental system and general solution, Wronskian: Abel's theorem, complex and repeated roots of the characteristic equations: reduction of order, inhomogeneous equations: methods of undetermined coefficients and variation of parameters, mechanical vibrations, forced oscillations and resonances.
- Laplace Transform: definition and main properties, solutions of initial value problems for second-order ODEs, ODEs with discontinuous forcing, impulse functions, convolution integrals.
- Systems of first-order ODEs: reduction of higher-order ODEs to system of first-order equations, Wronskian and independence, homogeneous systems of first-order, constant-coefficient, linear ODEs: general solutions, phase portraits for 2 X 2 systems (all cases: real, distinct, complex-conjugate, repeated eigenvalues), exponential of a matrix.
- Phase plane analysis: trajectories, critical points and their stability, locally linear systems, the pendulum.
- Partial differential equations: two-points boundary value problems for ODEs: eigenvalues and eigenfunctions, separation of variables, Fourier series: definition, Fourier coefficients, even and odd periodic extensions, heat conduction in a rod: fundamental solutions, solution by Fourier series, vibrating string: solution with zero initial velocity by Fourier series, D'Alembert formula for zero initial velocity.
© Anna Mazzucato Last modified: Wed April 27, 15:23 EDT 2011