Introduction to Slope Fields (Differential Equations 9)

hey we're almost done exploring what differential equations are and what the solutions look like we're almost done with that and we're gonna get moving on in the next few videos here like after three or four videos from now on the process of solving differential equations like what techniques we have and then we're gonna explore how to use those in applications and then some ideas of some different ways to solve them but for right now one idea that I want to share with you is is this idea of slope fields what if so field is why we might have some fields and this kind of breaking the pattern in our heads that we have to have an exact solution all the time in real life because that's not true so a lot of differential equations use approximations because there aren't any good techniques to solve some of these things we're going to come up against that's one of the ideas of slope field so in this video what I want to get at is what a slope field is how a slope field can help us identify a solution even if we can't explicitly define it and that's important so here's a slope field we'll go through a problem a solution ID in the process of how we're gonna go about it so here's the problem lots of times differential equations can be solved if they have this dependent variable Y in them you look at it over there's nothing we have it's not separable we'll talk about separable it's not linear talk about linear it's not exact it's not a substitution we can make it's not homogeneous well I don't understand those things so you're not supposed to right now but we're gonna get to those techniques of solving differential equations as we move forward so don't worry about it but the point right now is that sometimes we come across these difference of equations that have Y in them that we're not gonna be able to do anything with and you look at you go I there's nothing I can do so whatwhat's this possible solution to that that's what is so field and some other techniques helped us with so here's an example the derivative of Y with respect to X so a differential equation has a first derivative a first order differential equation as x squared plus 2y squared let me go that's got a Y in it there's nothing that we that's gonna help us to solve that and I there's nothing to do so sometimes you run into that in differential equations where there's nothing that we can do to get an exact answer an exact general solution or even a particular solution that's going to fit that so one solution to our problem is we can oftentimes approximate solutions and that works with for some practical purposes and a lot of people in math did when they get here they don't like that I don't like it I still don't really like it there's not a solution you know when we have linear equations you solve it and you get one answer or you get no solution or infinite solutions but we get an answer and there's a process to doing that well with the difference of the equation sometimes there's not and so we have to get kind of used to this idea that not all the time do we get the answer that we're looking for by a process that's outlined like step one step two step three it doesn't exist in some cases so we have these techniques to approximate and it works in real life really well so we say well this this well we can't find an actual solution we can approximate it it's gonna work just fine for us and if you remember we talked about that a few videos ago where we said that that these approximations are good enough that these approximations are something that that we can model life on and when we get to advance in our equations we can't solve them anyway so we have this trade-off if we start limiting variables well then we were not perfectly representing our situation but if we perfectly represent our situation then we can't solve it so sometimes in making an equation better we limit our ability to solve it and that's where approximations can come in and help us and say well I can't particularly solve this but it's good enough for the the real-life application of trying to represent so here's your idea so so again sometimes you can't solve it with the techniques I'm going to teach you here's a solution we have approximation so here's the idea on how to get that approximation one idea is called a slope field so here's an idea does if we know that first derivatives represent slopes but we know that a derivative is a rate of change a first derivative represents a slope of a function that we're dealing with so first derivatives represent slopes so if we have a first derivative a derivative of Y with respect to X derivative of the dependent variable and with respect to the independent variable if we have that solved for some function that has both X's and Y's in it what that means is we have a formula to find the slope that's kind of cool this represents the slope so if this is the first derivative this is the slope that's our formula so our first derivative our slope is represented by this formula with X's and Y's that's me so this means that we have a formula to find our slope more specifically this represent slope so the first rib to represent slope so we have a form of the final slope at any point at any point that we want to Y well why does it work well if this is our slope and it's based on X's and Y's points have an X into y it's going to x coordinate y coordinate by just plugging it into the function and that should be pretty straightforward to us if a first derivative means slope and this slope equals some function of x and y given any point both x and y we can plug that point into this function and find a slope at that point provided that it's continuous at that that point so provided that we were not an endpoint provided that we don't have a gap or a hole or it's not defined so that that's important we'll get into uniqueness and existence a little bit later two videos from now so short recap wrongs done I'm going to get to two examples to show you what slope fields are and they take a while to create they're not hard a lot of people use computer to do this or a calculator I'm not gonna be showing you that of course my ideas here are to get you to understand the concept not just to be able to do it but to understand what's going on and I like to do that by hand when I go through this so slope fields problem not we can't solve them all solution maybe we can approximate and one idea is hey if derivatives represent slopes and you have a differential equation with the derivative in it and we can solve it for a function of x and y then we have a formula to find slope at any point this is slope that's the formula just plug in the point to our function and you're gonna find the slope at whatever point x1 you just plug in so here's our process what we're gonna do is is we're gonna look at this we're gonna make a whole bunch of little lines that represent slopes at points on the XY plane it's not as hard as you might think you're gonna have this tons of slopes but there's a pattern to that I'm going to show you so make the graph so XY plane that has a lot of lines to represent little soaps so they look like unit vectors almost without the arrow fifty bad calc 3 and you have you had vectors so you see those little unit vectors all over the place they really look like that without the arrows at points on XY they just give a direction that your line is going well that's what slope does right it gives the rate of change at any point on a on a function and so our idea is if we make this graph of a whole bunch of little lines that say hey at this point the function should be going like this well then we can put all of that together and given a specific point in initial condition then we can find an approximate curve whose slopes fit that fit that feel at every point slopes fit and what we mean by that is it is this so here's the general idea in finishing vague because we haven't had any examples but here's the idea you're gonna take this function this differential equation that represents slope slope is a first derivative and you're gonna take every point on the XY plane and that sounds hard take every point and plug it in and say well at this point I get this slope at this point I get this look at this point I get this little and you're gonna make this huge XY plane there's gonna be slopes everywhere these little lines that represent direction at every point the slope can be given at any point just by plugging it in so with the whole XY plane we could potentially plug in every point carefully and every point but like the point three comma four point negative one comma two zero zero we can find the slope for any of those points where we have a defined function so where it's continuous where it's differentiable we can find the slope of so we're gonna do that we're going to take the whole XY plane we take every point on it that we can find then we're gonna find the slope at those points it's gonna look crazy we have all these little directions all these little slopes on this on this of this plane then if I give a specific point well we can say there's this is the way the curve has to go or has to travel to make it through that point and to fit all of these slopes so given a point or an initial condition that's what we're talking about then we can start approximating a curve so here's what you need to know about these slope fields that we're going to create when we create a slope field so all these slopes on the plane it's a general solution it says that this is everything that could happen they sort how all the curves would have to look to go through these these slopes to make sure that we're fitting the slope at every single if you want one specific herb in particular solution like we studied before you need an initial condition you need a point to go through so long story made short sometimes stuff don't work we need approximations one way is to find the slope at every point put it on a graph and then that represents your general solution that represents all the curves that would have such slopes that would have such slopes then what we do is we say if we want a specific one you just need a point and that will limit the curve so we'll find a particular solution that fits on our slope field that also goes through that point and at every point along that curve or fitting the slopes that's the idea well what I want to do now is show you two examples on how to create them and then all of this is going to make a lot more sense because as we go through they'll be explaining hey remember how this makes slope and remember how we need a point and now we can go ahead and see a specific curve so I'm gonna do that we'll do just two examples and I'll walk you through how to do that so let's get started on our example we're gonna build a slope field from scratch so no computers no calculators because I want to understand what's going on here they'll do it for you and I'm sure that you're gonna use it in whatever class you're in go and get a computer and that's the great way to build slope fields when you have some difficult differential equations so here's our problem again sometimes these things aren't solvable I'm not saying this one's not I'm saying I'm giving it easy example here so we can see what a slope field does so whether there's a technique or not to solve that explicitly I don't care right now what I want is your idea of slope fields to be built near it so you sure you understand how they work so here's our point if that right there means a slope then that's the formula for the slope so at every single point that we have X comma Y we can find the slope at that point of our general solution whatever curve is gonna fit this so our slope the first derivative is just X plus y that means that if we have our Y values and our X values we can make a table for slope really easily all we do is we have to add up our X plus or Y value so say take the x value + / it's going to give you the slope we're gonna make a table up like that it looks like it's gonna take a long time but there's a lot of good patterns here so we're going to I'm going to show you how to find the patterns I'm going to show you how to write it really quickly then we're gonna take our slopes over and put them on our XY plane and that will create a so field we'll talk about in just a minute now you can do this however you want I prefer to put my Y values of top of my X values on the side you can do it differently if you'd like to but the point is try to find a pattern and then and utilize that pattern so if you want to start here you can I might start looking for something like well if I'm if I'm adding up some numbers here I know that negative 4 plus 4 it's going to give me 0 are you seeing that 4 for my X plus negative fourth Y it's going to give me zero but so is 3 plus negative 3 and 2 plus negative 2 and 1 plus negative 1 and 0 plus 0 and negative 1 plus 1 and they did so all this diagonal is zeros the X values for the Y values negative 4 when you add them to find the slope which goes in this this field begin zeros all the way along that diagonal check you really want to but negative 4 plus 4 is going to 0 we can get the whole way now their pattern is gonna continue diagonally for this whole rest of it so if you find your first diagonal all you need to do is find one number and all these are gonna be the same so 3 plus negative 4 X plus y 3 plus negative 4 is negative 1 maybe try it for a couple of them so for these simple ones we get these nice diagonals do they all we're gonna bet no but a lot of them do and so we're gonna try that so negative 2 plus negative 3 is negative 1 also you probably could make patterns this way so 0 negative 1 that's probably negative 2 that's probably negative 3 that's probably negative 4 that's probably negative 5 we're excited way to make sure enough negative 4 plus negative 4 is negative 8 now we're going to continue our patterns all these my diagonals right now we've already got half of our slope field done that's pretty nice so if you're gonna have to do this by hand most of the time you can find some nice patterns if you can't use your computer now you this is just to give you an idea with a slope I'm not gonna explore that with you you can use a computer calculator to plug that in usually every single textbook that's out there has some sort of a graphing utility you can watch so through they're not bad check them out for yourself but this this is just to get the idea a little slope field does in your head same thing works here so if this is negative 1 0 that's probably one let's just check real quick 4 plus negative 3 that's one that's probably 2 it is so now what we've done we've taken this thing this differential equation which has a first derivative in it which means slope and we have a formula to find it we just plug in every single point for this finite graph that we have so it's it's limited but you could extend it as suppose and we find the slope by just adding the X and the y coordinates that's for this specific differential equation now we take all of these slopes and for point by point we're gonna put just a little line on this graph to represent the slope at this point it looks like a unit vector without the air isn't you giving you the direction that's what slope does gives you the instantaneous rate of change or the direction that the curve is going at that point what that's gonna do is fill out this entire XY plane with all these sort of slopes that represent your general solution all the curves that would fit these slopes so we'll get to that right now okay so now that we have this filled out all of our slopes at every point that we really need on the XY plane well we're going to put those little lines and every one of these points I tried to do this the first time without the the grid that's really hard it's gonna get all disorganized so make yourself a grid or did some seventh-grade paper some graphing paper that way you know every single one of those intersections of your X and your Y that's where you're gonna put your little line so when we do this the slope field what's happening is that at every single cross section of an X to the Y we now have a coarse fine slope so we just have to go along I'm gonna use my diagonals I'm going to start at four negative or I'm gonna make a diagonal so four negative four x equals 4 and y equals negative 4 x equals 4 y equals negative 4 I have a slope of 0 so if there's a particular solution that hits that point it is going to have a slope of 0 at that point is going to be flat so it's probably going to be like a relative Max or relative min it's gonna bounce somehow it's gonna do that now because you have these diagonals just follow your pattern at 3 on the x and negative 3 on the Y or Y on the X and negative 3 on the Y we're also going to the slope of 0 and we are gonna have that every one of these intersections hopefully you can verify by looking here that yeah you know what at negative 1 on the x and 1 on the line that you'd want to X 1 the Y of slope is 0 and a negative 2 to negative 3 3 and negative 4 for those all have a slope of 0 that's how slope field works and we're gonna take some time and just fill this out maybe you can do it as undo it if you've drawn this if not just follow along that's ok I have another one in just a minute that you can do in your own if you'd like so be careful it's kind of easy to get these confused especially confusing your X with Y or Y so I pause from for just a minute every time I start a new diagonal and I make absolutely sure that I have the right starting slope for whatever point I'm on so I'm gonna go and do 4 on the X negative 3 on the Y I'm going to put a positive slope of 1 so 4 on the X negative 3 in 1 for X negative 3 on the Y and that's gotta have a slope of 1 now I'm gonna see how that pattern works so the next one 3 on the X negative 2 on the wire 3 on the X negative 2 on the Y I'm gonna try to give that the same slope all the way up and drawing that agree it makes it infinitely easier but infinitely but a lot easier okay let's keep on going also if we can we can use this pattern down here so as we're going on the x equals or vertical lines so going up the Y's our slope goes from zero to one - two - three - four - five - six - seven - eight here's what that means this is a slope of zero one two three four five six seven eighths can be really steep so as we climb along our Y for a given x value our slope is increasing so you could do that - there's about a slope of two so I'm gonna continue that diagonal I have a slope of three four five six seven and eight and you're approximating here um so you're Schultz are gonna get really really steep by the end of it so that's stuff zero-one I followed it to I followed it I know this is going to be three zero one two three it looks pretty close here's going to be four so at 4 on the X and 0 on the y I better have said before on the X 0 why I've got a so before the next I should have the slope of five I'm going up on that x-value and my slope is increasing then 6 then 7 then 8 Malaysian hired novel so fuchsia this is looking like what this means is that every single one of our curves that fit on here at every point would have to have that slow so what how how many curves are there an infinite many that's why this represents a general solution so all of the solutions that we could possibly have would have to fit on these these this plane and have these slopes at their specific points that's some saying earlier that if you want to find a particular solution one curve you need to start with a point so we fill this out and then I'm gonna give you a point let me let me finish the rest of this and then we'll we'll go through and give you a particular initial condition to find a particular solution so let's finish the rest of our pattern at 3 on the X and negative 4 on the y we're gonna exit for the why I need episode with negative 1 at 2:00 on the X and negative 4 on the y so 2 on the x and negative 4 or why I need a silver negative to make sure that - 9 X negative 4 so to only acts negative 1 Y so by negative 2 then they get the 3 negative 4 negative 5 negative 6 negative 7 and negative 8 just like we have shown here about like that and we're approxima and going fairly quickly so let me recap it for you in just a bit some things we can't solve this this we can probably with another technique but we can find the slope at every point that so those so field does it says all of our X's overwise find the slope and every combination of points and then our coordinates for points then we put it on what's called a slope feel an XY plane now this is a picture of a general solution so this is every possible curve that would work for that differential equation it's gonna bid on here somehow and this is the way that they would look so to find a particular solution well you need to you need to have a point so let me show you a couple of things that we can do if we had a point like 0 0 say that we wanted to have the particular solution one curve that had an initial condition of y equal at Y of 0 equals 0 that would go through the point 0 for X 0 for y that's right here so if we're gonna have to go through that point remember what I said about every single point along this curve would have to have these slopes well we're backtracking now we're saying what if I need a curve that goes to that point well it's gonna still have to fit this look at that how this is this would have to fit along these slopes so if I'm going through that point yeah right here we're gonna have a change from decreasing to increasing we're gonna have a local minimum for this particular curve so right at that point we know that after that all my suits are climbing they're all positive they're all climbing before that all my soaps are negative they're all they're all falling is because that's right so this has to really be approximate by something that goes along increases look how this is gonna have this slope here goes further it's gonna have this slope it's gonna end with a fairly steep slope that has to look something like that to fit that slope field so once we are given a specific point in initial condition the rest of this flows along with their slope field now before that I know that that slope is zero it's gonna make it's going to dictate that we have this this change from decreasing to increasing and this line this line is all negative one right here like this is this later one slope says hey you're you've gotta follow that it's like an asymptote so before that we're going to have to somehow if it is and we probably looks something like that so general solution that's the whole thing this represents all possible solutions particular solution you just need a point and then follow your so field let's try a couple more how about how about negative 2 - so why I'm negative 2 equals negative 2 that gives us a point negative 2 comma negative 2 so let's go to that let's go to that point negative 2 negative 2 that's right here now we're not going to be hitting this asymptote because look how our slopes go I know that afterwards I'm I'm getting more and more negative with my slopes I have to look something like that before I'm just going to follow this up I'm going to follow my slopes I'm just going to make sure that I'm hitting I'm eating those slopes at though at the right the right slope for wherever I'm falling on my slope field so this was let's see a slope of negative 1 negative 2 negative 3 negative 4 so this would be negative 1 negative 2 so slope between negative 2 and negative 3 right there that's about how that particular solution work I hope this is making sense you I hope you're seeing the idea that the general solution has all these different slopes for a particularly at a point and just follow the slopes before and after that let's do one more of my y equal Y of 4 equals 1 so when we go to this point 4 comma 1 all it's saying is make sure that when you're when you get here at 4 comma 1 you're lying or your curve that you're gonna draw it's going to fit all the slopes before and after that so after that we have a slope of let's see 1 2 3 4 x over 5 we're going to have do this all that and it's going to be an increasing slope after that but before remember that we have this asymptote thing I'm trying to fit between my slopes here I know that from here to here I go a slope of 0 to 1 so I'm trying to make that pretty cool between there then somewhere in here I'm changing from this positive to a negative slope as I going backward so I'm going to hit a slope of 0 and then I have this asymptote that I got it and that gives us our particular solution that would have to go for that point it's pretty cool because we can model these approximations without doing any real hardcore math we know that our particular solution has to fit these slopes this little one so field tells you since it's got a figure now one more question what if I wanted to go through a point like 1 comma negative 2 1 comma negative 2 I am on that slope of negative 1 I'm not changing as I'm going down that so on this particular point before and after it we're just traveling along that asymptote and what that does it says yeah your if your slope is negative 1 as you move down that you're not changing you this is the slope is negative 1 I know we're going to show on specific points here but every point between here would also have a slope of negative 1 you can't get off of that that's the asymptote you're gonna eventually be on that the whole entire time so that's about it that that's how slope feels work we take this differential equation that we may or may not be able to solve and if we need to approximate it so if you don't have a technique to solve it we can create this field that has a whole bunch of slopes at different points then we just have to find the particular solution that goes through a point and fits that sub field so what you're getting when you do a so field is the general solution that's why their slopes everywhere you go well there's not one curve that goes through all of those of course is it won't even be a function to get a specific one particular function you need a particular point then we can match up from that point fit all the slopes before and after that so just notice that on the previous example I know we're only doing one today takes just take a long time and there's nothing different about any of these things that you do so this follow the same exact process just notice that how given that if you have a point a particular solution it gives you a unique solution to that differential equation all the paths through the slope field are our solutions to differential into that particular differential differential equation so all of all the different curves you can draw so every single one of them would be a particular solution of that differential equation put them all together you have the general solution I hope that makes sense to you the keys here are that we're approximating for things that we can't really solve so that's that's that's the idea practice a couple of them some simple ones by hand and then start using your computer or your calculator your graphing calculator to do harder ones I want to see for the next video where we'll do a couple more actual slope fields but I want to put them into practice I want to show you some real-life examples that way this isn't just really super Drive so see if next time for using some fields to solve some actual applications you

Channel: Professor Leonard

Views: 64,133

Rating: 4.9757943 out of 5

Keywords: Math, Differential Equations, Leonard, Slope Field, Equilibrium Solution

Id: m9Y8U9f9_Bw

Length: 34min 17sec (2057 seconds)

Published: Wed Oct 03 2018