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Problem 9 Hard Difficulty

In Problems $9-20$ , determine whether the equation is exact. If it is, then solve it.
$$(2 x y+3) d x+\left(x^{2}-1\right) d y=0$$


$C=x^{2} y+3 x-y$


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Video Transcript

in this video, we're gonna go through the answer to a question too benign from chapter 2.4. So ask to the servant whether the Given first order differential equation is exact, and then if it is to solve it Okay, let's first find out whether this equation is exact. Just do that. Let's find D m d y. Where m is the co vision off the X. So the, uh, the why derivative off this is gonna be just two. Why, it's two decks. I mean, because why only comes up in this first term that the end, the X where Ennis coefficient of the y it's gonna be well, the derivative of X squared is too exclusive. Month 10 These two are equal to each other, therefore is exact. It's not gonna solve it to solve it. We need to find the function f off X. Why? So we know that the ex derivative off at the expert intuitive off f is equal to em. So therefore, if we integrate that, if we integrate M, which is to X, why close three between straight that with this back to X, then we're going to get a function off. Why only. Okay, that's what that equation is saying. So, during this integral, uh, the two X is gonna integrate too. Two x squared over to the tunes of counsel this times by why the three is gonna become three X. I got the why function at the end. Okay. Said that four. We take the why derivative off this equation, then that's gonna be X squared. That plus the derivative Oh, G a y okay. And we know that the y derivative off F just by definition is gonna be equal to end x y, which was our coefficient of D y, which is with equal to X squared. Minus one. Okay, so now we have an equation with this guy and this guy, they're exquisite. Both sides. We can subtract from both sides. Therefore first derivative off G, why is equal to minus one? Therefore integrating the he's gonna give us g of why is equal to minus why? Plus some constant c. Okay, so now putting that back into the equation for F, which is this one we get of eggs? Why equals X squared? Why? Quest re x plus g of why? Which is minus y plus c And then we know that this that when effort is equal to a constant, we know that that is gonna implicitly define our solutions. So since we have a constant in here in G anyway, we put on the inside and it's smooth, constant, and call that c so he wants to see on, then the solutions are gonna be implicitly to find by this equation.