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# Exact equations intuition 2 (proofy)

## Video transcript

in the last video I introduced you to the idea of the chain rule with partial derivatives and we said well if I have a have a functions I Greek letters I it's a function of x and y and if I wanted to take the partial of this with respect to no I want to take the derivative not the partial the derivative of this with respect to X this is equal to the partial of Z with respect to X plus the partial of Z with respect to Y times dy DX and in the last video I didn't prove it to you but I hopefully gave you a little bit of intuition that you can believe me but maybe one day I'll prove it a little bit more rigorously but you can find proofs on on the web if if you are interested for the the chain rule with partial derivatives so let's put that aside and let's explore another property of partial derivatives and then we're ready to get the intuition behind exact equations because what you're going to find it's very it's fairly straightforward to solve exact equations but the intuition is a little bit more I don't want to say it's difficult because if you have the intuition you have it so what if I had to say the sames functions I and now we're to take the partial derivative of it of Z with respect to X first I'll just write Z I'll have to write x and y every time and then I were to take the partial derivative with respect to Y so just as a notation this you can write as you can kind of view it as you're multiplying the operators so it could be written like this the partial del squared times I or del Squared's I over del Y del or curly DX and that can also be written as and this is my preferred notation because it doesn't have all this extra junk everywhere you could just say well the partial we took the partial with respect to X first so this just means the partial of Z with respect to X and then we took the partial with respect to Y so that's one situation to consider what happens when we take the partial with respect to X and then Y so with respect to X you hold Y constant to get just partial with respect to X ignore the Y there and then you hold the X constant and you take the partial with respect to Y so what's the difference between that and with we were to switch the order so what happens if we were to undo it in a different color if we had Zhai and we were to take the partial with respect to Y first and then we were to take the partial with respect to X so just the notation just you're comfortable with it that would be so partial X partial Y and is it you know this is the operator and that's it might be a little confusing that here the between these two notations even though they're the same thing the order is mixed that's just because it's just a different way of thinking about it this says okay partial first with respect to X then Y this views it more is the operators that we took the partial of X first and then we took Y like you're multiplying the operators but anyway so this can also be written as the partial of Y with respect to X sorry the partial of Y and then we took the partial of that with respect to X now I'm going to tell you right now that if each of the first partials are continuous and most of the functions we've dealt with in the in you know in a normal domain as long as there aren't any discontinuities or holes or something strange in the function definition they usually are continuous and especially in a first-year calculus or differential course or we're probably going to be dealing with continuous functions in in our domain if both of these functions are continuous both of the first partials are continuous then these two are going to be equal to each other so Z of XY is going to be equal to Z of Y X now we can use this knowledge which is the chain rule using partial derivatives and this knowledge to now solve a certain class of differential equations first order differential equations called exact equations and what does an exact equation look like an exact equation looks like this well it's always the color picking is a hard part so let's say I have this is my differential equation I have some function of x and y so you know I don't know it could be x squared times cosine of Y or something I don't know it could be any function of x and y plus some function of x and y we'll call that n times dy DX dy DX is equal to zero this is well I don't know it's an exact equation yet but if you saw something of this form you're your first impulse should be Oh what will you actually refer very first impulse is is this separable and you should try to play around with the algebra a little bit and see if it's separable because that's always the most straightforward way if it's not separable but you can still put in this form you say hey is it in the exact equation and what's an exact equation well look immediately this pattern right here this pattern right here looks an awful lot like this pattern right what if M was the partial of Z with respect to X what if Z with respect to X is equal to M right what if this was I with respect to X and what if this was I with respect to Y so Y with respect to Y is equal to n what if I'm just saying we don't know for sure right if you just if you just see this someplace randomly you won't know for sure that this is the partial of with respect to X of some function and this is the partial with respect to Y is some function but we're just saying what if if this were true then we could rewrite this as the partial of Z with respect to X plus the partial of Z with respect to Y times dy DX equal to 0 and this right here the left side right there that's the same thing as this right this is just the derivative of Z with respect to X using the partial derivative chain rule so you could rewrite it you could rewrite this is just the derivative of Z with respect to X and Z as a function of X Y is equal to zero so if if we look at you see a differential equation it has this form and you're saying boy that you know I can't separate it but maybe it's an exact equation and frankly if you know that that was what was recently covered in you know in a before the current exam it probably is an exact equation but if you see this form you say boy maybe it's an exact equation if it is an exact equation and I'll show you how to test it in a second using this information then this can be written as the derivative of some function zai right where this is the partial of Z with respect to X this is the partial of Z with respect to Y and then if you could write it like this and you take the derivative of both sides are you take the antiderivative of both sides and you would get zai of XY is equal to CE as a solution so there are two things that we should be caring about that you might be saying okay Sal you've you've kind of you know you've walked through XY's and partials and all this what is it what how do I know that it's an exact equation and then if it is an exact equation which tells us that there is some zai then how do I solve for the sine so the way to figure out is it an exact equation is to use this information right here we know that if sy and its derivatives are continuous over some domain that when you take the partial with respect to X and then Y that's the same thing as taking as doing it in the other order so we said this is the partial with respect to X right this is the partial with respect to X and this is the partial with respect to Y so if this is an exact equation if this is the exact equation if we were to take the partial of this with respect to Y right if we were to say if we were to take the partial of M with respect to Y so the partial of Y with respect to X is equal to M if we were to take the partial of those with respect to Y so we could just rewrite that as that then that should be equal to the partial of n with respect to X right the partial of Z with respect to Y is equal to n so if we take the partial with respect to X of both of these take the partial with respect to X we know from this that these should be equal if Y and its partials are continuous over that domain so then this will be this will also be equal so that is actually the test to test if this is an exact equation so let me rewrite all of that again and summarize it a little bit so if you see something of the form M of X y plus n of X Y times dy/dx is equal to zero and then you take the partial derivative of M with respect to Y and then you take the partial derivative of n with respect to X and they are equal to each other then and it's actually if and only if it goes both way this is an exact equation an exact differential equation this is an exact equation and if it's an exact equation that tells us that there exists as I such that the derivative of Y of XY is equal to 0 or is I of XY is equal to C is a solution of this equation and the partial derivative of Z with respect to X is equal to M and the partial derivative of Z with respect to Y is equal to n and I'll show you in the next video how to actually use this information to solve for is I so here are some things I want to point out this is going to be the partial derivative of Z with respect to X but when we take the kind of exact test we take it with respect to Y because we want to get that mixed derivative similarly this is going to be the partial derivative of Z with respect to Y but when we do the test we take the partial of it with respect to X so we get that mixed derivative so this is with respect to Y and then with respect to X so you get this anyway I know that might be a little bit involved but if you understood everything I did I think you'll have the intuition behind why the methodology of exact equations works I will see you in the next video where we will actually solve some exact equations see you soon