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26. dy +> = (ry)", y(1) = 4 dx 4 (2xy? + y) dx (2y' _ x) dy = 0. 8. (Sx - y + 1) dx (6x 2y - 3) dy = 0. 16. Find family of oblique trajectories that i...

Question

26. dy +> = (ry)", y(1) = 4 dx 4 (2xy? + y) dx (2y' _ x) dy = 0. 8. (Sx - y + 1) dx (6x 2y - 3) dy = 0. 16. Find family of oblique trajectories that intersect the family of circles x? 2 at angle 45".

26. dy +> = (ry)", y(1) = 4 dx 4 (2xy? + y) dx (2y' _ x) dy = 0. 8. (Sx - y + 1) dx (6x 2y - 3) dy = 0. 16. Find family of oblique trajectories that intersect the family of circles x? 2 at angle 45".



Answers

Find a family of oblique trajectories that intersect the family of circles $x^{2}+y^{2}=$ $c^{2}$ at angle $45^{\circ}$.

In this problem were given a family of curves defined by the equation. X squared plus y squared equals two c x. Our goal is to determine what differential equation do we need to discover in order to find the orthogonal trajectories to that curve. There were to determine that differential equation. We first must differentiate our original equation with respect to X, and that differentiation must be done implicitly, since the dependent variable Y is to a power of two. Let's start off. On the left hand side, the derivative of X squared becomes two x. The driven of white scored will give us plus two y times D y the X equals, and now we're ready to take the derivative of the right hand side. The independent variable is X has only variable in the right hand side, so he obtained just a to C circle. Next is to solve for D Y DX. We'll start off by first subtracting two extra both sides. So now to why Times D Y T X is equal to two C minus two X. Then we could take is our next step to divide both sides of this equation by two y If we do that, we obtained that. D Y T X is equal to si divide by why minus X divide by Why, since the factor of two cancels throughout as our next step, we need to note that the constant C depends on both Excel y at the same time due to the original equation. So it's taken extra step to rewrite this equation. X squared plus y squared equals two c x and then solve it for the variable. See, we divide both sides by two x we obtained that see is now equal to X squared plus y squared divide by two X, and that substitution needs to be made here. Let's go back to this original derivative for why it cannot be expressed as D y DX equals one over why times The expression for C and C is X squared, plus y sward divide by two x so that finishes the substitution. And let's just copy a negative X over why The next thing we ought to work with on this equation is to find its most simplest form. With between the two fractions, we can simplify by including a factor of two in the numerator denominator as well as a factor of X. So put in the denominator, too, as well as X numerator. We have to and multiply by X. Then this would simplify too. X squared plus y squared divide by two x y from the first fraction minus two X squared Divide by two ex wife from the second fraction. If we subtract across the numerator, we'll obtain that the numerator is y squared minus X squared. Divide by two x y every prefer X squared. Coming first weaken. Factor out a negative sign from the numerator to obtain negative of X squared minus y squared. Divide by two X y is the slope to the family of equations. So now we're ready to state our solution. The answer to this problem is that the differential equation is found by writing de y over DX and taking the negative reciprocal off the last expression we just found. So our differential equation will read d Y d X equals positive two X y divide by X squared minus y squared. So that solves this problem

All right. So we'll be using the same method. That's what we did in problem for, if you to during the orthogonal trajectories or the family of circles even scrape this. Weisberg all's OK. So first that's different. Treat, this is music. So we get the dean over the X of our first term expert plus e over the X off our second term. Weiss Weird and me over the X off our third term on the art of term. On the right hand side. Just a script. Okay, so the derivative of X rays to X and then the derivative of y squared still Why me? Why Over the X okay. And the derivative off Our constant expert is just okay then solving for the went over the X recent track both sides by two x So we have to Why, b y over the X equals negative two x Hey! And then we divide both sides by two y teas and so this one out and simplify we have the y over the X. Okay, because negative X mover wife today and the family off the trajectories orthogonal to are given family of circles have despise is that the diva. The slope s negative. Reciprocal off this one. Okay, so the negative off X over y iss X over y and then it's reciprocal. Eas y over x. Therefore, we have the why over the X because white race. Okay, then finally they get the general solution of our slope off the trajectories. Arthur vinals, Our families are safe health. So using variable separation, we separate. Our variables were in all of the wives who have been left hand side. All of this is right. So we have b y over y calls the X over X. Okay. Then get the anti derivatives. Both sides we have integral of B y over why it's out. Enough. The absolute value. Why? Because the angle of the X over X at the selling of the absolute value of X and the arbitrary constant C three close. Sorry. Yeah. Okay, so we get the power's off your boots sites we have. Why? Because X times the recent. Okay, so since series to see, use a positive constant. Our solution This can be retain. Asked why support to pay extra in case? Yes, The Okay,

So for this problem we are interested to find out the equation of attention to an ellipse with equation for X squared plus Y squared is equal to 72. Even that the tangent that we are looking for is perpendicular to the line with equation X plus two, Y plus three is equal to zero. So considering the line with this equation, we can actually write this in the stuff in there said form and that should be equal to negative one half, X minus the origin. And from this form we could say that the slope of the line is negative point half. If the tangent that we are looking for is perpendicular to this, then the slope of the tangent to be equal to the negative reciprocal of the slope of that line. That being said, it should be equal to the negative of The -1 over -1/2 which is the slope of the line. So the slope of the tangent should be equal to two. So, this is the first thing we know. Now considering these ellipse for X squared Just why it's quite possible to 72. We can actually generate an expression describing its the slope of the tangent. Do any of the points on these lips. So that should be eight x. DX. That's two. Y. T Y is equal to zero and Levi over TX is able to negative eight x over To buy or simply -4 X over Y. Okay, and this is the slope of the tangent. Any tangent towards that. Arabs. Now, I know the stop of the tangent that I'm looking for? It's equal to two based on this description that we have that it is perpendicular to that line. So let me say empty is equal to -4 X over two. Sorry to Y. Oh let me just correct this. I kind of message here so dy over dx here is negative four X over Y. So the slope of the tangent is negative four X over Y. And this negative four X over Y is equal to two. The tangent of them interested to find the equation is having a slope of two. So I'll just simply equate this With two. So from here we can somewhat create an expression saying -4 X is equal to do why or simply why is equal to negative 2X. Now I will use this relationship defined the actual point of tangent C. Point of tendency. Okay, so from this equation for X squared, That's why it's pretty simple to 72. I know that why is able to negative two x at the point of the tendency. So we say four x this is native to X. Screen people to 70 so we'll end up having yeah eight X. Symbol 72 or X squared is equal to nine Excess Pressure -3. Okay, now I know what access I can talk for the value of Y. Let's take all that from here. It's just why it's about the -2 x. So that being said when taxes three, Why is 6 -6? When X is -3 why is positive six? So the point of intersection is a three comma negative six. In a -306. So I know the stone. I know the points. I can I can definitely solve for the equation of the night. Using this equation Why minus price of one is equal to m times X minus X. A point. So for the first point will have y plus six Is equal to two times X -3. So we'll end up having two X. Okay so let's have it. In the sloping slope intercept form we have Y is equal to two X two X -20. The other tangent line will have this equation using the second point of tendency would have Why minus six is equal to two times X plus three Or why is able to two x. Kristen In these two here are the two equation of the tangent lines to these ellipse That is perpendicular to the line. Express to Wipo's 30-0

Well, here we are solving the push Number 30 We have given X squared life. Boy square? Yes, they called stool. I don't see a way that's differently. They questions implicitly with respect to X, which will be two x less so. Why the life? My DX is because you to see. Do you like baby? It's becomes the wife. Yes, X My guess why market as a question one? This contains constant C So it is not the difference immigration off the given family so we can express the value off X on the bill in question. But this this one so you can read it is see is a question exclude? No. Why? Square divided by the way. Now substituting this an equation one you get Oh, well, why my two years? Yes, he called Stool x. A group? Yes. Why Square do Why? I mean, it's a tropical store X squared plus y squared to square Modify boy, which becomes door. That's why stroll square minus my square. From this equation, we get the difference Still equation for the family of portable tragic story. This is the wife ideas The coastal my Nestle exclude minus y script divided by dual X y. Can the witness do you? Why my idea? There's pickles toe minus one minus. Why buy Hex? We go. Do you know why? Thanks now Substrate puting. Why speak with stool? Thanks. Be thanks. You convey d y by D it? Yes. Equals to be less x b by the X we've close. Do you be by D X close to one, minus this square by Do we? No, thanks. Nick will be on the right side. Next. Davies idea. Nicholas, stool, maybe square my next one after it so operated form you will be by three b squared minus one. This sheik was still give me a yes. It goes to the X by X now in to get both the sites of the question left side by B and they would say by X like the some instigation will be three square in this one. They request to integration one by eggs in your smart get is in question Took so the integral off this side it is the left side. Let's put previous governments one is equals to think which we come six me Do you me some cold stew about into girls becomes intimidation. My B Dave Days Stool The Natural Law Day. Let's see. Stop shooting back. The value for these questions do becomes three. Natural law No. Three Squared minus one. Is it still natural? Next messy that comes through this gloominess. One. We'll kill. We'll still see one. Knicks No substituting back or B is a cold too. Live by s. We get dream. Why buy eggs? Hold square minus one. My cue Because do anyone necks? No, this is cool. 23 way square. My news It's good. Hope you close to see line to the bar for So this is a question of the family of portable and tragic stories. Oh, next let's drew it. This is 100 another 100 No y Axis 200. This 100 someone to use Bring one for the family of what I wanna trajectories for the oneness because to 90 which way? Who and these and I'm raising that one for six was 2 20 True. Yeah. Move this. No


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