Exam reviews
Major topics include:
Verifying solutions to ODEs; solving initial value problems by solving for appropriate constants; finding general form for solutions by superposition
Methods for solving first order ODEs:
Separable (separate; integrate; uncomplicate)
Linear first-order (integrating factor)
Exact (find "potential" function)
Substitution methods (including homogeneous, Bernoulli, order reduction, etc.)
Autonomous ODEs (y'=f(y)) including classifying critical points, drawing phase diagrams, drawing some typical solutions, and description of long-term behavior from initial conditions
Using the Wronskian (e.g. to test linear independence)
Solving constant coefficient linear homogenous ODEs by using auxiliary polynomial, including cases involving repeated and complex roots
Solving nonhomogeneous ODEs using the method of undetermined coefficients or the method of variation of parameters
Using ODEs to model/solve application problems such as mixing (e.g. tanks), population (particularly the logistic model), and springs (including classifying as undamped, underdamped, critically damped, and overdamped)
Major topics include:
Modeling using systems (e.g. mixing with multiple tanks; systems of springs)
Solving systems of equations by substitution
Going from a high-order differential equation to a first-order system (by introduction of new variables); going from a linear system to a high-order differential equation (method of elimination / variant of Cramer's rule)
Using the "D" notation
Carrying out basics of matrix algebra (scaling, addition, multiplication, inverses)
Rewriting systems of equations in matrix form, and vice-versa
Verifying first-order systems
Using the Wronskian (e.g. test linear independence)
Finding solutions, both general and with initial conditions, of linear systems using the eigenvalue method including situations with complex and repeated eigenvalues
Finding the fundamental matrix for linear systems; finding exponentials of matrices (including special case of repeated eigenvalues)
Using method of undetermined coefficients or variation of parameters to solve nonhomogeneous ODE systems
Major topics include:
Knowing and using the definition of the Laplace transform
Taking Laplace and inverse Laplace transforms by use of a table
Rewriting expressions, particularly partial fractions, to help take Laplace and inverse Laplace
Handling piece-wise functions with the unit step function; using the delta function
Solving differential equations by use of Laplace transform including applications (e.g. springs)
Solving systems of differential equations by use of Laplace transform
Using properties of Laplace transforms, including shifts, derivatives, and convolution (Duhamel's principle)
Manipulating power series (including shifting indices, pulling off terms, combining, multiplying, ...)
Working with recurrences of coefficients
Finding the first few terms of a series solution to an ODE
Finding two linearly independent solutions using power series
Using power series to find solutions to ODEs
Finding radius of convergence; distance to singular points