Chapter 1 Introductory Concepts

Sec 1.3 – Solution Curves & Slope Fields  IOW,  Recall that the solution y is just a continuous function  So it has a curve (graph) just like any function  What’s more, the equation itself tells us a lot about the function  Its derivative y’ is given by f(x,y)  So the slope of the solution curve at (x,y) is given by f(x,y)  So in a tiny area around (x,y), we know what the graph of y looks like.

Example  1.  Note what this says about the slope of the solution function y at any point  Slope at (1,2)=  Slope at (-2,1)  Slope at any point where y = -x is

We can sketch these easily  y’ = x + y

Nothing special about x, y being integers  y’ = x + y  For example, at (.5,.5), m tan =  And at (-.5,0),

We can use as many mesh points as we like  y’ = x + y  The solution curves start to stand out

How do these relate to particular solutions?  We know that y is smooth  For this reason we can assume it connects the little slope marks in a smooth, natural way (see previous page)  Each of these curves is a particular solution, corresponding to a particular IC (more than one, actually)

Let’s see if actual solutions match  Solutions to the DE y’ = x + y are where C depends on the IC  Each C generates one of the curves on the picture.  Suppose y(0)=-1. Then -1 = Ce 0, so C = 0, and the solution function is y = -1-x  One of the curves on the graph should be -1-x

Let’s try some other C’s  Plot for C = 1, 2, 3,.5,.1, -2, and 0 on same TI screen

Example – Sec 1.3 #13  Plot  You can now do the problems from sec 1.3

Plotting with technology  Plotting a DE on the Nspire CAS  Plotting a DE in Maple