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Problem 7 Easy Difficulty

In Problems $1-8,$ classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.
$$[2 x+y \cos (x y)] d x+[x \cos (x y)-2 y] d y=0$$

Answer

The equation is not separable, not linear, so the equation is exact only.

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Video Transcript

in this video, we're gonna go through the answer to question number seven from chapter 2.4 so as to show whether this equation inseparable, linear or exact or none of these. Okay, firstly, is it separable? It's quite clearly not because you condone you're just never gonna be able to split up this cause x y or this calls ex wife into a function of X times. By a function of why you just never gonna be able to see that So is not separable. Is it linear again? It's not linear because cause of X Why is a nominee a function of X on dhe? A moment of your wife is exact. Well, let's see. Let's try and work out What gm the wise when M is this guy? Eso differentiate that inspector why we used the, uh, the product rule for the second term. Okay, so that's gonna be because x by close. Why? Times x AA minus at times my minus sign specs. Why? Okay, so let's just go through that. So, firstly, we different rate the at the y on left cause X y alone. Then we left. Why alone on different trade cause X Y ah, we do the chain with your friend shake cause that's why that's why we bring the X out the front and then caused different shades to my side. Okay, so this is cause thanks. Why? Plus rooms minus next. Why sine x y Then let's see whether that's equal toe the end The X Where is the commission Off the d y in the equation. So again, using chain rule? Yeah, it's gonna be cause text. Why minus, Let's see X times Why times sine x y that's using the chain will once again Okay, so the M d y is equal to the MDX, so therefore, it is an exact defense in question.