00:01
All right, let's have some fun with differential equations.
00:04
First thing we're going to do is find basically our equilibrium points, figure out if they're stable or unstable, and draw some graphs.
00:11
So let's have fun with this.
00:13
So our first problem is dyvt equals ay plus b y squared.
00:19
We are told that a is greater than zero and also b greater than zero and that we're interested in looking basically for y being from minus infinity to infinity.
00:34
All right.
00:34
So to solve this, what we're doing is we're going to look for equilibrium points.
00:37
That's when the derivative goes to zero.
00:40
So we're going to go ahead and factor out y.
00:43
That will give me a plus b, y equals zero.
00:48
So therefore, we're looking at equilibrium points at y equals zero.
00:52
And let's see, we've got to solve a plus by equals zero.
00:57
By equals minus a, y equals minus a over b all right so these are our two equilibrium points so i like to actually do my phase line next but we will draw a couple of the graphs besides the phase line okay so we have zero and minus a over b i know those are to the left because both a and b are positive and so we are looking at values of y and then what d y d t looks like these are zeros all right so what we we can do is into our derivative, we can plug in a really big value.
01:32
We'll definitely get a plus here.
01:34
If we plug in something in between zero and minus a over b, we'll get a minus sign.
01:42
And we plug in very large numbers.
01:43
We get back to positive.
01:45
So what's happening here is that i diverge away from zero from both sides, but i will converge towards minus a over b.
01:54
So this is our stable node at minus a over a.
01:59
Whoa, we gotta write that right.
02:01
So stable, this is our stable equilogram point or stable node, and this is our unstable node.
02:12
Okay, if we wanted to do a quick, we have a couple more graphs real quick.
02:16
One is just a y as, sorry, dy, dt as a function of y.
02:23
And we can see we have a parabola and both a and b are positive, so it opens upward.
02:29
We know our zeros, are at zero and over here minus a over b.
02:36
So we get something generally that looks like this.
02:40
So that is our basically our graph of the derivative function with respect to y.
02:46
But now we're going to come up with basically solution curves.
02:52
And we have vertical asymptotes.
02:56
Here let's let's redraw that little bit because i want to have room with the vertical asymptotes.
03:01
All right.
03:03
So i guess i'll just do the vertical astinototes in a different color so we can see what's going on.
03:07
Okay, so here we are going to graph basically now we're going to graph y, dy, d, t as a function of time.
03:24
Okay, actually sorry, y as a function of time.
03:26
Let's get this right.
03:27
Okay.
03:29
Okay.
03:30
So let's see.
03:31
We know that.
03:32
When y is zero, we have a horizontal.
03:39
Basically, that's kind of a horizontal asymptote.
03:43
And when y is minus b over a.
03:48
So we have that as well.
03:50
Oh, minus a over b.
03:51
So minus a over b and here is zero...