Learning goals

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,

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.

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.

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 Handouts 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 (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.

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 $\rightarrow f(x)$)

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).

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