MATH295

Classify DEs as to type, linearity, and order

Use various notational forms for derivatives

Understand the distinction between explicit and implicit solutions

Understand the distinction between an initial value problem and a boundary value problem

Understand families of solutions, particular solutions, and trivial solutions

Understand and use piece-wise defined functions, and their notation

Understand and use functions defined as integrals, and their notation

Understand the distinction between the existence and the uniqueness of a solution

Find intervals and/or regions where unique solutions must exist

Use calculus and algebra to solve for multiple unknown coefficients or parameters

Understand and use direction fields

Understand, write, use, and solve autonomous DEs

Find critical points and construct phase-portraits of autonomous DEs

Find and classify constant solutions to DEs

Rewrite and solve separable DEs

Find the domain of a particular solution as compared to the domain of a family of solutions

Find integrating factors and understand their application to first-order linear DEs

Use anti-differentiation and tables of integrals to solve first-order linear DEs

Write solutions to first-order linear DEs that involve unknown integrals

Understand constant solutions, increasing solutions, decreasing solutions, and logistic solutions

Identify and interpret transient and non-transient solutions

Use partial derivatives and partial anti-differentiation to solve exact DEs

Understand t-homogeneity and perform various substitutions

Solve Bernoulli DEs, including applications

Write and solve DEs that model linear situations in physics, engineering, and population dynamics

Write and solve DEs that model non-linear situations in physics, engineering, and population dynamics

Use Kirchhoffs law to model and solve problems involving LRC-series circuits

Use systems of linear DEs to solve problems in physics, engineering, and population dynamics

Understand higher-order DEs and fundamental solutions

Work with the linear dependence and independence of a set of functions and compute Wronskians

Understand the difference between homogeneous and non-homogeneous higher-order DEs

Understand and utilize the superposition principles of homogeneous and non-homogeneous higher-order DEs

Use reduction of order to find a second solutions to a given second-order DE

Solve homogenous linear DEs with constant coefficients, up to a reasonable order

Express all solutions to a homogenous linear DE with constant (and real) coefficients as real functions

Solve for all unknown coefficients in a solution, given initial conditions or boundary values

Solve non-homogenous linear DEs with constant coefficients, up to order five, by using the superposition approach

Solve for all coefficients of the particular solution to a non-homogenous linear DE with constant coefficients

Work with annihilators, including their compositions

Solve non-homogenous linear DEs with constant coefficients by using the annihilator approach

Use variation of parameters (Lagranges method) to solve second- and third-order linear non-homogeneous DEs

Use annihilators and elimination to solve systems of linear DEs

Solve for relationships between the coefficients in systems of linear DEs

Rewrite unknown functions, derivatives, and simple products as power series

Re-index and combine power series

Solve recurrence relations for a reasonable number of iterations

Use power series to find solutions to homogeneous and non-homogeneous DEs about ordinary points

Recognize DEs and power series that result in special functions, such as the Gamma function

Understand functional transforms, specifically the LaPlace transform

Find the LaPlace transform of basic functions by hand and use tables for more complicated functions

Rewrite piece-wise-functions using unit functions and find their LaPlace transform

Discuss functions for which the LaPlace transform is defined and undefined

Solve partial fractions problems with up to eight unknowns and find inverse LaPlace transforms

Find LaPlace transforms of derivatives using initial conditions

Use LaPlace transforms to solve DEs with continuous and piece-wise defined solutions

Write systems of DEs using vector notation and verify solutions to a system of DEs, using vector notation

Use Wronskians to determine whether given sets of function vectors are linearly dependent or linearly independent

Find all Eigenvalues (including complex) and associated Eigenvectors for a homogenous system of DEs

Use vector and matrix notation to solve homogenous and non-homogenous systems of DEs