Difference between revisions of "Learning goals"

Revision as of 13:55, 25 January 2017

Midterm 1

General knowledge of differential equations

• Classify equations by linearity, order, ordinary/partial, non/homogeneous.
• Definitions: solution, general solution, particular solution.
• Verify that a given function is a solution to a given ODE.
• Determine arbitrary constants using ICs.

First order equations

• Determine correct integrating factor and solve equation.
• Plot integral curves of an ODE (whole family of functions parametrized by an arbitrary constant).
• Determine how many possibilities are there for $\lim_{t\to\infty} x(t)$ for different ICs, including parameter dependence.
• Solve separable equations. Implicit form - pick the correct branch for given IC.
• Modeling:
  • Given a word description, write down ODE(s), e.g. non-homogeneous Newton's Law of Cooling,
  • Given a word description, write down ODE(s), where description is, for example, of inflow/outflow to/from one-tank or unidirectional two-tank problems, possibly with time dependent inflow and/or total volume,

Second order equations

• Determine independence of functions using the Wronskian.
• Determine dependence of functions by finding a linear combination that sums to zero.
• Work with complex numbers and Euler's equation.
• Solve second order equations - solve characteristic equation and translate into general solution: distinct real roots, repeated real root, complex roots.
• Given form of ODE and the general solution, determine the details of the ODE.
• Use reduction of order to find a second solution when a first solution is given.
• Recognize the structure of solutions - where is homogeneous part + nonhomogeneous part reflected in solutions?
• Method of undetermined coefficients (MUC). ODEs with any RHS that has a finite family (or, by the end of term, Fourier series) and cases in which RHS (or member of its family) solves the homogeneous equation.
• Determine correct form of $y_p$ for MUC (or work backwards to figure out unspecified RHS from given general solution).
• Determine unknown coefficients in proposed $y_p$ as per MUC.

Midterm 2

Mass-spring systems

• Rewriting a sum of sin and cos as a single phase-shifted cos (or sin).
• Write down ODE given mass-spring physical parameters (mass, damping coefficient, spring constant, forcing function).
• Determine over/under/critical damping and how parameters influence this classification.
• Vibrations / forced mass-spring systems.
  • No damping - beats, resonance, response amplitude plot.
  • With damping - practical resonance, response amplitude plot.

Systems of ODEs

• Finding eigenvalues, eigenvectors and generalized eigenvectors.
• Writing down the general solution to systems with (i) real distinct eigenvalues, (ii) repeated real eigenvalues with a complete set of eigenvectors, (iii) repeated real eigenvalues without a complete set of eigenvectors (i.e. a defective matrix), (iv) complex eigenvalues.
• Find steady states (constant solutions).
• Classify steady states using trace/det plane, parametric curves in the trace/det plane, identify crossings (changes in classification at specific parameter values). Note: there are no problems of precisely this type in webwork; look at the midterms posted on the Solutions page.
• Relate vector field to system of equations, vector field to solution.
• Two-tank problems, including non-homogeneous term (constant vector case only).

Laplace transforms for solving second order equations

• Laplace transforms (calculating them from the definition)
• Using partial fraction decomposition for Laplace inversion.
• Completing the square for Laplace inversion.
• Writing piecewise linear functions in terms of Heaviside-type functions ($u_c(t)$).
• Solving IVPs involving various forcing functions (exp, trig, polynomial, piecewise) using the Laplace transform approach.
• Delta function properties.

Post-midterm 2

Laplace transforms for solving second order equations

• Modelling with the delta function.
• Solving IVPs with delta function RHS.
• Transfer functions, impulse response and convolution.

Fourier series

• Calculating FS.
• Using FS to solve second order ODEs with periodic forcing (Method of Undetermined Coefficients)
• FS for even and odd functions.
• FS convergence (for what values of x does the FS $\rightarrow f(x)$)

Heat / Diffusion equation

• Flux $J_a$, conservation of mass.
• Steady state solutions.
• Using Fourier series to solve the Heat/Diffusion equation.
  • Homogeneous Dirichlet BCs (zero concentration).
  • Homogeneous Neumann (no flux).
  • Non-homogeneous Dirichlet BCs (fixed concentration).
  • Non-homogeneous Neumann (fixed flux).