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Find the general solution of the differential equation. $sqrt{1-4 x^{2}} y^{prime}=x$...

Question

Find the general solution of the differential equation. $sqrt{1-4 x^{2}} y^{prime}=x$

Find the general solution of the differential equation. $sqrt{1-4 x^{2}} y^{prime}=x$



Answers

Find the particular solution to the differential equation $y^{\prime}\left(1-x^{2}\right)=1+y$ that passes through $(0,-2),$ given that $y=C \frac{\sqrt{x+1}}{\sqrt{1-x}}-1$ is a general solution.

All right. So we have with the French allocation expert. Why? Prying equals quantity experts one times white. So what we're going to do is we're right. Why? Prying into the form we y over the X. We have x squared me. Why? Over the X Because quantity X plus one se slash Okay, now we'll separate our valuables all off the bounds are all of the terms with why will be on one side and off except the right side. So let's identify all of the terms with one when we have this one, then all of therms with X have experts one an expert, Kate. So by default, will, uh, we'll use a loo. Consulted it all the terms with X on the right hand side because existed in on an otherwise the left hand side a the dividing both sides by why we have b y over y. And then you have the X on the other side. Divide both sides. My expert, we have X plus one over X squared the X. Okay, Now we get the anti Guerry with their both sites or get ah, the anti girl outside to have b y over y and X plus one over. Expert gets, But we can actually our signified the right hand side. We have grill off one of bricks plus x rays, too. Negative, too. Yes. Hey. And you see addition property. And he goes, we have a little boy. Absolute. While they're flying balls to go off one over X. Is Helen off the actual value of X plus, the eagle are X rays negative to its negative. One of bricks. Uh, here we have negative one over the X. Sorry. Should be mean. Yeah, negative. One of Rex. And then don't forget the plus. Okay. And then will now solve for white. Okay, So, solving for why can action get the e off old sites? But before that, before that will are each this one? You have Ellen off the options by the white Bulls, Ellen of the actual of X minus small over x. Hey. And then get with the exponential forms off both sides. Hey, so we have, uh why? Because you re still Ln of X plus one over x This here and using the loss off or the exploitation moves girls and have exponents. This could be reading us here east to L A X Thank you, Reza Wall over X times he raced. Okay. And then simplify Mia. Why off X series one over X. And then that's the fine Meurice to see. Ask Capital Letter A Hey, science, we arranging we have boy WASPy x e Recent one over X. Okay, so this is over. What? Because they are our generals.

All right, so we have differential Asian white. Frank was two x y squared. Hey, first thing is that's right. Wife Prime me. Why? Over GX? So this is the why over the X boss to ex White Square. Okay, And then that's new variable separation. That's with all the weiss on left hand side and all the excess on the right hand side. Hey, so we have the y divide both sides by y skirt. Do you have anyone over? Why square and thats were attained to x on the right hands. Idea to x e x. Yeah. So now our valuables are already separated. We'll get the answer derivative off both sides. Fusion the anti the riveting off the Y over y square s negative one her y and then we have the anti derivative off two x dx. This is two X squared over two. And then don't forget the constant they are carry constant c. Okay, so you can cancel this one out to here, and so will have negative one over. Why? Because X clear this. Okay, then. Well, now, Saul for why can actually be across mobile application or multiply. Why on both sides and in divide expert Percy on both sides. And we'll simply get why all snagged one over X squared, plus c. Okay, so this is our general. Okay, so is this one?

Okay, so let's start with, uh, separating our variables so we can divide both sides by t in squared off. I minus one minus y square squared off tee times one minus y squared. Then multiply both sides Spite. Hey, so pencil this one out. Hansel Bt and then cancel this the leftward B y over a squared off one minus y square. Because Bt over. Okay. And then get the integral off boots sites we have Indy girl off the Y over squared off one lines. Weiss. We're which is actually equal to, uh are exciting. Where? The universal sign. Inverse sign. So this is our sign. But, boy, because Elden off the absolute value of team and then don't forget the arbitrary constant. Okay. No way. You have to Seoul for our white. Okay, so get the sign off boots sites yet the same yet the same function. Yeah, will have. Why? Because sign false held in off the absolute values team. Let's see. Okay, so this can be considered this hour. General solution. Okay,

Hello, everyone. Today we're going to sort problem. Number 15 here Therefore for Mexico toe X in the root off one minus X squared F effects Have nice X can wilderness eggs in the root off one minus X squared. Differentiating which can be done as X dish in the root. Tough one minus X square plus X window wrote off one minus X squared. The whole life prove it can grate on us root of one minus X squared, plus eggs in the day by delay off Route one minus eight Square, which is a Gatto wrote off one minus X squared, plus eggs in Do the by the way off, one minus X squared in the root off my minus X squared figured like Rudolph one minus X squared plus X Hindu debate The window one minus X squared in your debate, one minus X square in the root off, one minus X square. So why do they get things like fruit off one minus X squared minus to work the square in the one by two in the road off one minus X squared difficult toe off one minus X squared minus X squared. Different way, wrote off one minus X squared, Very difficult. One minus X squared, divided by where it off one minus X squared, minus X squared, divided by square root off one minus X squared friend minus two x squared They were. They square root off, one minus x squared. Thank you.


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