DEFINITIONS, THEOREMS, AND PROBLEMS

TO KNOW FOR MIDTERM 1

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

     Differential equation

     Solution to a differential equation

     Initial value problem and solution

     Separable first order o.d.e.

 * Autonomous o.d.e.; equilibrium solution; stable, semi-stable and unstable

     Exact o.d.e.

Bifurcation point

* Picard Iterates

Principle of Induction

 * Euler’s Method (define the iterates and explain with a picture how the method works)

Homogeneous linear differential equation, inhomogeneous linear differential equation

     Wronskian of two functions

Linear dependence/independence of two functions

THEOREMS     Be prepared to prove *’d theorems

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

      Existence and Uniqueness theorem for first order differential equations

 * Exact equations theorem (Theorem 2.6.1)

   Existence and uniqueness theorem for linear second order IVPs

* Principle of superposition

 * Two solutions form a basis for the solution space of a second order linear differential equation if their Wronskian is nonzero (Theorems 3.2.3 and 3.2.4 together)

 *  Abel’s Theorem

TYPES OF PROBLEMS  (not an exhaustive list)

     Show that a given function solves a given differential equation or 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

     Construct the first few Picard iterates for a given first order equation

     Approximate solutions to IVPs using Euler’s method with only a few steps by hand

     Show two functions form a set of fundamental solutions for a given linear 2 nd order equation

     Find a pair of fundamental solutions for any second order homogeneous linear equation with constant coefficients with distinct real roots.

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.