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O Consider the differential equation$$\frac{d i}{d t}+a i=b$$where $a$ and $b$ are constants. By drawing the slope fields corresponding to various values of $a$ and $b,$ formulate a conjecture regarding the value of$$\lim _{t \rightarrow \infty} i(t)$$
As $t \rightarrow \infty$ the solution curves approach b
Calculus 2 / BC
Chapter 1
First-Order Differential Equations
Section 3
The Geometry of First-Order Differential Equations
Differential Equations
Harvey Mudd College
University of Michigan - Ann Arbor
University of Nottingham
Lectures
13:37
A differential equation is…
33:32
01:04
(a) Make a conjecture abou…
02:00
Slope Field Use the slope …
02:44
Use a computer algebra sys…
00:43
10:35
Consider the initial-value…
07:57
Prove that $\lim _{x \righ…
04:13
Let $y_{1}(t)=t^{2}$ and $…
07:27
Consider the differential …
02:03
(a) Use a CAS to graph the…
02:52
A differential equation A …
04:20
Sketch the solution of $\d…
00:38
Find the limits. Write $\i…
01:14
Show that if $$F_{\mathrm{…
05:03
Sketch the slope field of …
02:01
Consider $P(D) y=0 .$ What…
04:42
Verify that the given func…
03:40
Second-order linear differ…
04:06
Use a Taylor polynomial to…
03:56
It can be shown that solut…
All right, so we're looking at I prime. Plus, we're looking at this differential equation for constants A and B A, Nizar function eyes. The function that we're looking at A and B are constants, and we're looking at this differential equation. Now. What we want to know is, what what is the behavior of? I, as I goes has said that the variable like t say goes to and hey, Andi, uh, so I mean one we can we can make a very simple conjecture about this. Which is that my ghost to be over in a, um because thea equilibrium solution to this is be over a, um if you plug in B over a this is B. This is zero. So this is just be, um eso It would make sense for the behavior solutions to go towards that equilibrium. Now, if we draw the slow field, we hope that it reinforces this conjecture. So it this is why? Because e on, uh And then this is, um why access and X axis. And then after that, we think about, say, the slope field. At this point at this point, why is larger than be OK, right Yeah, or you say I I you know, that's just because we're thinking about it is a curve I as a function of xer. Really? What we should be doing is thinking of this, like as the I access and this is a T axis, and this would be I equals B over A And this would be some point with I larger than pure ivory. So then the minus ai, that's negative. Yeah, this that's negative. So but this is equal to I prime, which is negative. So this goes down by exactly the amount de and the finest ai this. And since this really only depends on I so we can do something like this, but it's really just just go down more and more. Shockley. The farther away you get from this line a similar argument by a similar argument. If I as larger wait if I a smaller than be over a and being minus I, which is I prime, is larger than zero. So this is increasing really, really steep. They almost look radical and so slow feels look like this and occur for this Looks like that. Yes. And so basically, as as t goes from tyranny. As we go out really far along the T axis, I approaches his equilibrium. Value B equals a so we can conjecture that this limit is be over a
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