Problem 39 Hard Difficulty

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
$$
\lim _{t \rightarrow \infty} i(t)
$$

Answer

As $t \rightarrow \infty$ the solution curves approach b

Related Courses

Calculus 2 / BC

Differential Equations and Linear Algebra

Chapter 1

First-Order Differential Equations

Section 3

The Geometry of First-Order Differential Equations

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Video Transcript

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

Get More Help with this Textbook