Lastly, we can use these pictures to sketch

the general form of solutions to this differential equation.

So here's a different equation: Newton's Law of Cooling,

- that specifies the derivative - how the temperature changes,

as a function of temperature, and here's a plot of the right hand side,

the same thing I should before, that shows how the rate of change

in Celsius per minute - degrees Celsius per minute,

depends on the temperature, and here's the phase line

- we have a stable, attracting fixed by at 20,

and we can use this to sketch solutions T(t).

So this phase line is similar to the phase line for iterated functions,

now I'll sketch some solutions that are similar to the time series plots for iterated functions.

Let me draw some axes... So here are my axes,

and now on the horizontal axis is time - t in minutes,

and then here's temperature - degrees Celsius,

So I know that, - let's say, do this point in purple,

if my starting value is 5 - so I'm going to start somewhere here,

I know I'll increase until I get 20 - I know I'm going to start at a rapid rate of increase

because this function has a large value - the rate of cooling is large,

and the rate of cooling gets smaller and smaller, as I approach 20.

So, I don't know the exact details but I know the curve has to look something like this:

I'm going to approach the stable fixed point at 20,

and initially, I'm warming up very quickly - this function is large

and then the rate at which I'm warming up decreases as I approach 20.

I could do another solution - suppose we started with a different beverage

at, say, 45 degrees - then I would cool off very rapidly

- this is very large and negative so I'm losing temperature quickly,

and this might look something like that.

So, I can't from this qualitative picture figure out the exact functional form,

or the exact timing of this - I'll show you how to do that in the next lecture,

but we can get an awful lot of information.

So, this is called a qualitative analysis of a differential equation,

- we sketch the right hand side, and we see where is the function increasing,

and where is it decreasing - it's increasing whenever the derivative is positive,

it's decreasing whenever the derivative is negative.

From there, we can immediately draw a phase line,

and from that phase line we can sketch the general shape of solutions.

So, I think this is a lot of information just from a little bit of geometry and common sense

and very often this will be enough to analyse a differential equation.