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Numerade Educator



Problem 1 Medium Difficulty

A direction field for the differential equation $y^{\prime}=x \cos \pi y$ is shown.
(a) Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) $y(0)=0$
(ii) $y(0)=0.5$
(iii) $y(0)=1$
(iv) $y(0)=1.6$
(b) Find all the equilibrium solutions.


b) $$y^{\prime}=x \cos \pi y$$


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Video Transcript

were given a differential equation and its slope field and in part a your given initial conditions and were asked drone. The solutions containing these initial conditions we see in part by our solution has an extradition. Y zero equals zero. This tells us that if I draw this solution in black or graph has a point at 00 and staying this tangent to the slopes as possible, we have a teen. The graph for one we see that it's symmetric about the line X equal zero. It looks something like a co sign curve that doesn't repeat itself. L. Griff two in red. You see, that initial solution is why zero equals 00.5. So we have a point at 0.5 if you follow the slopes as tangents to our curves, we see that we obtain a straight line, which is why equals 0.5. So I draws three in green. See that we have initial tradition. Why it zero equals one. This tells us that we've a point at 01 and staying tangent to the slopes as much as possible. The obtained another curve. When we see that this curve is identical to one except it's reflected about the line y equals 0.5. Finally, play graph for in black again. We see that initial condition is y zero equals 1.6. So 1.6 lies on this cur. And therefore, if we follow the slopes as tangents for our curve, get teen curve, which approaches infinity as X approaches positive or negative infinity and it's symmetrical about the line X equals zero in Part B were asked to find all equilibrium. Solutions call that an equilibrium solution is a value where we have some function. Why such that? Why prime is equal to zero? So this would imply that's X co sign of pie. Why is equal to zero and because why prime is equal to zero? We have that why is equal to see some constant in the rial members. So therefore, this is equal to expo sign of pie See And since this is euro for all values of X, including non zero values of X, it has to follow that co sign of high C is equal to zero. Therefore, hi c must be equal to hi over to plus and pie are some end in the temperatures. What this tells us is that in fact, our function number four from part A doesn't go to infinity. Instead, we can expect that it will flatten out near the top. Yes, we have occurred more like this and this curve will be approaching what we had before Hi over to is approximately. Okay, so we need to finish solving for C So supplies that c is equal to 1/2 plus and over pie and is an element of the interviews. So we have seasonal 1/2 and we can see this clearly. We also have it. C is equal to 1/2 plus one over pi, which is about 1/2 plus 1/3 is going to be the present pie. So that should actually be 1/2 plus end. So seize, you gotta wanna have placentas. So we have 1/2. We have three hats which we see is also the acid toe. And we should have another ass in tow at why equals £5. So this is the ass in Toto, which are fourth. Kurt approaches as experts, plus and minus infinity and five hats is going to be greater than two. But less than 3 22.5 So 2.5 is going to be somewhere right about the end of the y axis drawn. So we're getting these. This curve is actually going to perch infinity as X approaches plus and minus infinity. So we see that equilibrium solutions are going to be functions. Why equals 1/2 Plus in where and his and elements of the intruders I'll call each one of these. Why's that, then? This is our answer.