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University of Missouri - Columbia

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Problem 13 Medium Difficulty

Match the differential equations with the solution graphs labeled I-IV. Give reasons for your choices.

(a) $ y^{'} = 1 + x^{2} + y^{2} $

(b) $ y^{'} = xe^{-x^2 -y^2} $

(c) $ y^{'} = \frac {1} {1 + e^{{x^2} + {y^2}}} $

(d) $ y^{'} = \sin (xy) \cos (xy) $

Answer

(a) Graph III
(b) Graph I
(c) Option IV
(d) II since y is always $0,$ no matter what x equals, $x(0)$ will always equal $0 .$ which means $\sin (x 0)$ will always be $0 .$ Which means that $\sin (x y) \cos (x y)=0 \cos (x y)=0$

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Video Transcript

Okay, So for this question, we're asked to match the differential equation. What's grabbing the pollution? We're gonna do this by seeing what properties in the graph we can deduce from looking at the differential equation. And we're going to do this by noticing that why prime should be greater change of whatever is represented by the graph or, if you prefer the slope of the line tension to the attendant of curve of the point. Okay, so for a first of all, we know that why prime is going to be always greater venter equal to one, because I swear, plus y squared is always greater than an equal to zero. So the only solution to this differential equation needs to be strictly increasing. That narrows our options down to either three or four to decide between the two. We know that expert plus y squared gets bigger as you get further away from the origins that's both X rayed and watch where decreases. Remember with Morgan. So you would expect that the slope of this this graph for the rate of change would increase as you get further away from urgent, which leaves us, which is three as our solution for a So a gets matched with Bree for B, we now know ah eat too. That native X squared plus y squared is always greater than zero, but axe changes sign as you flip over the organ so e to the negative X Square, plus what are so The solution to this function should be increasing on one side and decreasing out the other. Which leaves us with either one or two. Two. However, to cannot be a solution to be because if we assume why is equal to zero identically, then why prime and B is not also identically equal to zero since we're just left with Why Prime People's X times on the night of X squared, which is not a constant function As a result, the solution to be cannot be constant and at least if y if there are can't be constant. Um I cant be wife will here are you So we're left with just one as an option for B okay for sea once again eating a negative each of the X square plus y square is greater than zero Are you not greater than zero? Why prime is going to be greater than zero everywhere. So we have a strictly increasing function again. Even so, just career forest option. Since we are in history, we can assume this is for like if we don't know that A was three already, we could deduce a seat was for from the fact that the denominator is of the faction increases the move away from border. So why Prime gets smaller and smaller as you mean forever further away from Oregon. So you would expect the rate of change to decrease, which is exactly what happens, and for now, for D our only auction off this too. However, you can also see that if you plug in y equals zero into the differential equation for deed. Yet Y prime equals sign of zero times Coast on zero. So why prime is equal to zero. So the constant function is a solution to the constant function. Y equals zero is a solution to be

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