Question
Exercise 6.3.7: Solvex(t) = cOS t +cos(t - t)x(r) dr:
Exercise 6.3.7: Solve x(t) = cOS t + cos(t - t)x(r) dr:
Answers
In Exercises $71-76,$ find all the solutions of the equation. $$\cos t=-1$$
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In this question Here. What we call that on Tiny Revert, You on the side the way. Go to the miners. Go inside the let's see answer entirely riveted on the course I d d t we go to the side D. C in this question were given the d S r d t ego Judy. Of course, I've day minus society on the initial value as off the pie equal to one. And here the first thing you need to find the ass it's illegal to the integral of the course I t minus society d d And then we can split into the two Integral. Now, of course, I t. And then Manus integral on the side day. Yeah, using the second formula here on down the river them the course I echo to the society on Tiny Review on the side Using this one, you get into to the plus the grow site de and then plays a constant See here on then. Now this will be the ass now on were given the initial value here. So we have asked on the pie. It will go to the side on the pie plus goes on the pie. Let's see. So every computers on this unequal to the zero this one will be minus one. Let's see way get equal to the one here. So it means that's it. Will Ego Thio see if we equal June the value to here? As a result, began the as off the assembly were equal to the society Plus go Zaidi plus you And this will be the anti knee revert you were looking for here.
We want to solve the given differential equation, which is dy dx is equal to the coastline of X divided by y minus one. This question is challenging our ability to find solutions of differential equations in particular. It's challenging us to use our newfound method called separation of variables. This method requires three steps I've listed below to solve the differential equation. First we'll isolate X and Y as well. Put Y terms the last X terms in the right. Then we'll integrate both sides of the equation since each side of the equation of the differential, then we'll solve for the integral to the integration methods to land in our solution. So first isolating X and Y gives us why minus one do Y is equal to the coastline of X. Dx. Thus, if we integrate both sides we have internal y minus one. Do Y equals integral. Co 60 X. Now we use integration methods to solve. Doing so we have one half Y squared minus Y is equal to the sine of X plus C. The constant of integration.
Who we want to solve this equation. The coastline of T. Is equal to zero. And we know that on the unit circle that the X coordinate is our co sine of the angle. So our angle is going to be T. And we can see that up here, we have a point on the unit circle at 01, and down here we have a point at zero negative one. So these are our two locations where we have the co sign of that angle on this will rotate positively. So that could be angle T. And that could be angle T. So T can equal pi over two Plus two Pi K. Or T can equal three pi over two Plus two Pi K. And that is where K is a member of the images. You may need to write that in words, I'm writing in symbols.