1.1 Direction field Vector (Direction): velocity, acceleration (magnitude and direction) Scalar: weight, height, speed, mass (only with magnitude)

Diff Eq. : dy/dt = f(t,y) 1.

y=y(t) --> unknown A.f --> given B.

Aim: Find solutions y=y(t) 2. significance of a direction field:

A solution of (1) is a function V=v(t) whose graph is a curve in the tv plane each short line segment is a tangent line to one solution curve in Ex.1 , if v 0 --> v(t) is increasing with time Threshold value/ critical value would be 49 because dv/dt =0 v(t) = 49 is a solution of (1) called the equilibrium solution

Equilibrium solution (of a diff eq.) : is a constant solution For the diff eq. dy/dt = f(t,y) (2)

f --> rate function RK (remark): the direction field is constructed by evaluating f at each point of a rectangular grid

Conclusion : lim y(t) =3/2 for any solution y=y(t)

t--> infinity

Day 2 Def: a differential equation is an equation containing the derivatives of a function

Def. : an O.D.E. is called linear if it is of the form ... Def: An Initial Value Problem is an O.D.E together with some initial conditions, which specify a particular solution Main Task in this course:

Given an O.D.E , does a solution exist? 1.If a solution exists, How Many are there? 2.

How can we find all solutions, if one exists? 3.