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Good day students will come two-part one of my slope fields uhm installment in this tutorial was going to be going over how to determine the equation of the differential equation given a slope or direction field okay so the instructions are as follows determining the differential equation that could be represented the given slope fields. For number one we have the sum circular slope field right here in the question is a which of these differential equation our results in this slope field okay right so we are not the whole idea behind the slope field we have a differential equation unity so we differential equation that C component gives you a family of curves that our party solutions of the differential equation so this slope is basically tells is the slope of the tangent line at each point on the family of curves that represents the solution to the differential equation the one after she the slope field to particular solution we have to find the initial value okay right so let's some see what how we can determining which of these equations can be represented by the Psalm slope field so the quickest way to designate do this using methods of elimination so for Syria going to do is take a look at the slopes the slope field these are all tangent lines and we ask yourself cardiol identical in here all identical that tells you that the slope field is independent of the X or Y values it's a constant slope field in that case your derivative are your differential equation is going to be equal to the constants right so if you look at this of the field use noted that the all the same the have the same steepness so to speak of this of this differed and this is see the steepness is decreasing and it hits zero and then assertion goes into the negative of area answer can see that it's not constants right since it's not constants we know that's the why the X equals a constant cannot be answer so the can eliminate option a option a would have been to beef the entire slope field had identical so okay for steepness right now that would then we down with eliminated one optionally down to four over going to do is we're going to fix one variable and ask ourselves how the other variable impacts the steepness of the slope field okay so let's some fix our X how will fix our why first isolate six are why is user red one here in red right so of if I wanted you speak this region up here and the take this entire region why equals to I'm going to fix my why and when asked myself the following question with wife fixed and X very in dollars the steepness of the line change view these as tangent line sections of tangent lines okay to the steepness of this tangent line change in the answer is absolutely yes so since it's changing that means that X is going to be present in the differential equation okay right let's do of take a look at the other orientation okay so the other orientation the going to look at Tom look at fixing the of X the sixty X at X equals positive to for example okay X equals published to see what that looks like that when the job start six right here okay now look at these these of the slope lines right here this tangent lines okay if we hold X fixed right and we allow white to attain different values does the alteration of the variability of the why Valley impacts the steepness of the line the answer is yes you noted that the slope of each line is different right here is undefined here is negative and of negative and here its positive decreasing the steepness so to speak okay so is increasingly increasing gets undefined in then he gets a negative in the subsequent is increasing again so of this means that the why is going to be present in the differential equation okay so says we, after doing are two investigations we know that in the differential equation why and X are present the must be present in the differential equation because the vote impact the steepness of the line will going to do that our out alternative and ask ourselves which of these ones of has year X or Y a long okay and the answer is option be option B has only and X okay so this cannot be a possible solution because X and Y impact play a role in this has only X this equation is telling us that of why doesn't zero only X three zero but in our meeting investigation just now we found out that both variables play a role okay right now we have three options that's easy for us to determining which one the answer is so what we do is the examine not slope field and we can look at the extreme cases for example this is an extreme case right here what is the slope of this line this as an undefined tangent line okay have an undefined tangent line acts when why is zero so that points that point right there is one, zero so when why zero we notice that we have the vertical tangents final undefined slope so a look at all this to three candidate
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