Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer.
$ y' = \sin x \sin y $
So for this, probably want to match the differential equation with its direction. Field onda differential equation we're given is y equals or why Prime equal sign ax Sign y on. The reason why we want to be able to do this is to recognize how, um, graphs play a role in functions. How functions can be a mapping of a graph on when we look in real life, we don't always have just a nice equation. We often have graphs and and data points, so it's nice to be able to represent that in terms of these kinds of graphs so we can perform calculations. So it's important that we're able to recognize what graph this belongs to. Um, and what we end up seeing is that this is the graph of to the reason we know that is because the slope of the line segments should be zero when X or Y is an integer multiple of pie. So consider the fact that the sign of pie two pi three pi four pi five pi All those, um, sign value of that is going to be zero. So what we end up seeing is that the whole column should be the same when X is one of those values and the whole row, uh, will be Why, if it equals one of them. So what we see is that the graph to meets that description because the slope, um, is at zero, nearly at zero when you have an X value of saying negative three or positive three because it's very close to the value of pi 3.14 So that's why we know that that's the graph to