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# Exact equations example 3

## Video transcript

welcome back I'm just trying to show you as many examples of possible of solving differential equations exact differential equations one trying to figure out whether the equations are exact and then if you know they're exact how do you figure out the Zhai and figure out the solution of the differential equation so the next one in my book is 3x squared minus 2xy plus 2 times DX plus 6y squared minus x squared plus 3 times dy is equal to 0 so just the way it was written this isn't superficially in that form that we want right what's the form that we want we want some function of x and y plus another function of x and y times y prime or dy/dx is equal to 0 we're close how could we get this equation into this form well just divide both sides of this equation by DX right and then we get 3x squared minus 2xy plus 2 we're dividing by D X so that Dix just becomes a 1 plus 6y squared minus x squared plus 3 and then we're dividing by DX so that becomes dy DX is equal to what's 0 divided by DX well it's just 0 and there we have it we have written this in the form that we need in this form and now we need to prove to ourselves that this is an exact equation so let's do that so what's the partial of M this is the M function right this is the M function what's the this was a plus here what's the partial of this with respect to Y this would be 0 this would be minus 2x and then this the 2 so that's the partial of this with respect to Y is minus 2x what's the partial of n with respect to X this would be 0 this would be minus 2x so there you have it the partial of M with respect to Y is equal to the partial of n with respect to X and y is equal to and X so we are dealing with an exact equation so now we have to find zai the partial of Z with respect to X is equal to M which is equal to 3x squared minus 2xy plus 2 take the antiderivative with respect to X on both sides and you get Z is equal to X to the third minus this x squared Y because Y is just a constant plus 2x plus some function of Y right because we know Z is a function of x and y so when you take a derivative with this when you take a partial with respect to just X a pure function of just Y would get law so it's like the constant when we took when we first learned taking anti derivatives and now to figure out zai we just have to solve for H of Y and how do we do that well let's take the partial of Z with respect to Y and that's going to be equal to this right here so the partial of Z with respect to Y this is 0 this is minus x squared so it's minus x squared this is 0 plus h prime of y is going to be equal to what that's going to be equal to our n of X Y it's going to be equal to this and then we can solve for this so that's going to be equal to 6 y squared minus x squared plus 3 can add x squared to both sides to get rid of this and this and then we're left with H prime of y is equal to 6y squared plus 3 anti derivative so H of Y is equal to when is this 2y cubed plus 3 y and you could put a plus C there but the plus C merges later on when we solve the differential equation so you don't have to worry about it too much so what is our functions I I'll write it in a new color our functions I as a function of X and why is equal to X to the third minus x squared y plus 2x plus h of Y which which we just solve for so H of Y is plus 2y to the third plus 3y and then there could be a plus C there but you'll see that it doesn't matter much actually I want to do something a little bit different I'm not just going to chug through the problem I want to kind of go back to the intuition because I don't want this to be completely mechanical let me just show you what the derivative using what we know and you before you even learned anything about that before you learn anything about a partial the partial derivative chain rule what is the derivative of Y with respect to X what is the derivative of Z with respect to X here we just use our implicit differentiation skills so the derivative of this I'll do it in a new color it's 3x squared minus now we're going to have to use a chain rule here so the derivative of the first expression with respect to X is is well let me just put the minus sign and I can put like that so it's 2x times y times the first plus the first function x squared times the derivative of the second function with respect to X well that's just Y prime right it's the derivative of Y with respect to Y is 1 times the derivative of Y with respect to X which is just Y prime fair enough plus driven this respect to X is easy to plus the derivative of this with respect to X well let's take the derivative of this with respect to Y first we're just doing implicit differentiation in the chain rule so this is this is plus 6y squared and then we using the chain rule so we took the derivative with respect to Y and then you have to multiply that times the derivative of Y with respect to X which is just Y prime plus derivative of this with respect to Y is three times just doing the chain rule the derivative of Y with respect to X that's Y Prime let's try to see if let's see if we can simplify this so we get this is equal to 3x squared minus 2x y plus 2 so that's this term this term and this term plus let's say let's take all the wife let's just put the Y Prime outside y prime x so you have a negative sign out here minus x squared plus 6y squared plus 3 so this is the derivative of this of RSI as we solved it and notice look at this closely and notice that that is the same hopefully it's the same as our original problem what was our original problem that we started working with the original problem was 3x squared minus 2xy plus 2 plus 6 y squared minus x squared plus 3 times y prime is equal to 0 so this was our original problem and notice that the derivative of Z with respect to X just using implicit differentiation is exactly this so hopefully this gives you a little intuition of why we can just rewrite this equation as the derivative with respect to a with respect to X of Zhai which is a function of x and y is equal to 0 because this is the derivative of Z with respect to X I wrote it out here it's the same thing it's right here right so that equals 0 so if we take the antiderivative of both sides we know that the solution of this differential equation is that is I of X and y is equal to C is the solution and we know what sy is so we just set that equal to C and we have the implicit we have a solution to the differential equation all those defined implicitly so the solution you don't have to do this every time this step right here this step right here you wouldn't have to do if you taking a test unless the teacher explicitly asked for it I just wanted to kind of make sure that you know what you're doing that you're not just doing things completely mechanically that you really see that the derivative of Zhai really does give you like we solved for zai and I just want to show that the derivative of Z with respect to X just using implicit differentiation and our standard chain rule actually gives you the left-hand side of the differential equation which was our original problem and then that's how we know that the derivative of Z with respect to X is equal to zero because our original differential equation was equal to zero you take the antiderivative both sides of this you get the derivatives you get zai is equal to CE is the solution of the differential equation or if you wanted to write it out zai is this thing our solution to the differential equation is X to the third minus x squared y plus 2x plus 2y to the third plus three Y is equal to C is the implicitly defined solution of our original differential equation anyway I've run out of time again I will see you in the next video