DEFINITIONS, THEOREMS, AND PROBLEMS

TO KNOW FOR FINAL EXAM

DEFINITIONS

Be prepared to explain *’d definitions in a short essay.

Up to midterm

     Differential equation

     Solution to a differential equation

     Initial value problem and solution

     Separable first order o.d.e.

     Exact o.d.e.

Linear 1st order ODE, homogeneous and nonhomogeneous

* Picard Iterates

 * Autonomous ODE, equilibrium solution, stable, unstable, semistable solutions

General first order and general second order ODE

Homogeneous linear second order differential equation, inhomogeneous

linear differential equation

     Wronskian of two functions

Since midterm

Natural frequency of a spring system

Quasi-frequency of a damped spring system

Quasi-periodic, Critically damped, and overdamped spring

Systems

Homogeneous linear system of equations

Vector space

Subspace

Span

Linear independence

Basis and dimension

Identity matrix

Inverse matrix

Determinant of an nxn matrix

Eigenvalue/eigenvector of an nxn matrix

Characteristic equation of a matrix

Wronskian of a set of n vector valued functions from R-->Rⁿ

THEOREMS

Be prepared to prove *’d theorems

Before midterm

 * General formula for the solution to a first order linear equation

with constant coefficients

 * Exact equations theorem (Theorem 2.6.1)

Existence and uniqueness theorem for general first order ODEs

    Existence and uniqueness theorem for linear second order IVPs

 * Principle of superposition

Since midterm

Properties 1-9 of matrices in section 7-2

Theorem 3, packet page 143

* In any vector space V the zero vector is unique.

* The eigenvalues of a matrix M are the roots of the equation

det (M - lI) = 0.

 *  Subspace theorem (Theorem 4 p.145 packet)

* Matrix multiplication distributes across matrix addition (Prove in 2x2 case only)

Theorem on equivalent conditions for nxn matrix invertibility (from class Wed 11/16)

Theorem 7.4.1 (solutions of a linear homogeneous system form

a subspace of the space of vector valued functions)

Theorem 7.4.2 (The solution space of a system of n equations is

n dimensional)

TYPES OF PROBLEMS  (not an exhaustive list)

Before midterm

     Show that a given function solves a given differential equation

or system or related IVP

     Draw a slope field diagram for a given differential equation

     Solve any linear, separable, or exact first order o.d.e.

     Set up differential equations models given information about

rates or forces

     Do word problems like the basic ones in section 2.3

Show two functions form a set of fundamental solutions for a

given linear 2nd order equation

Solve a constant coefficient second order linear homogeneous ode with real roots

Since midterm

Find a pair of fundamental solutions for any second order

homogeneous linear equation with constant coefficients.

Use the method of undetermined coefficients to solve

nonhomogeneous second order linear equation.

Solve undamped unforced and damped unforced spring

problems

Set up systems of linear equations based on mixing problems

Solve matrix equations using Gaussian Elimination

Determine if a given set of vectors is linearly independent

Determine if a given vector is in the span of a given set of vectors

Find the determinant of an nxn matrix

Find the inverse of a matrix up to 3x3

Find the eigenvalues and eigenvectors of a matrix up to 3x3

Solve 2x2 constant coefficient homogeneous linear systems

Draw the phase plane for any 2x2 const. coeff. hom. lin. system

A Word to the Wise on Calculational Problems:

Do the problems that were assigned not to be turned in to practice for calculational problems. Problems like these WILL APPEAR on the exam.