In each of the Exercises 1 to 5, form a differential equation representing the given family of curves by eliminating arbitrary constants $a$ and $b$.
$y^{2}=a\left(b^{2}-x^{2}\right)$

Answer

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Calculus 2 / BC

NCERT Class 12 Part 2 - Math

Chapter 9

Differential Equations

Section 3

General and Particular Solutions of a Differential

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Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Video Transcript

well no form the differential equation for this given family. Of course equation that is y squared equal to eight times of b squared minus x squared. A few things that we need to remember before we find the differential equation is uh differentiate the family. Of course as many as many times as the number of uh arbitrary constant. So if there are about to arbitrary constants we have to differentiate the equation two times. And in the next step what we do is we eliminate the already constants using the equation of why, why prime or Y double prime exit job. We can do this using elevation or substitution method. So this is this next thing. And once we have done this will be left with differential equation which is free after arbiter concern which is the required differential equation. So let's start doing the steps one x 1 first. What we do is we observe harmony, arbitrary constants are there in this family of course equation, observed that there are about to arbitrary constant which is A. And then they'll be So let's uh differentiate this couple of times. Let's differentiate this once. So when we differentiate the left hand side of this equation, differentiation of y squared is too white uh white prime because we have to apply the changes over here. When you apply the changes, we have to apply the power rule first that is differentiation of y squared is too white. And then we have to differentiate why? Which is why printing. So that's why I returned to y Y prime and this side we have this E As it is, this is a constant and we have to differentiate the inside part of it differentiation of B squared which is basically a constant, that will be zero And then differentiation of X. Queries two weeks. So we write down like this so therefore we can read on this as two. Y Y prime is equal to eight times self -2 works. So notice that we can cancel the common factor in both ways. That is too, we can cancel this too just like that. And when we do that we'll be left with A Y. Y. Prime is equal to negative X. Then uh we just differentiate at once. We will have to differentiate this one more time. So let's differentiate this one more time. They're going to differentiate this equation. Notice that this is a product involving Y and Y. Prime. So we have to apply the product rule. So when we apply the product to take the first term that is as it is. And differentiators again to which is the white prime differentiation of why primaries, Y double prime plus. Now take the second term of cities that is why prime and multiply with the differentiation of the first term, that is differentiation of Y is y prank. So this is equal to next to a Claims differentiation of excess one. So we have the situation that simplified this. We have why Y double prime plus Y. Prime squared Is equal to -8. So So we consider this is the equation one I forgot to know right on this and this is the equation to. When you compare these two equations we have to solve for it. So let's take this post equation. I'm going to take this post equation why why prime Is equal to 92 x. A console A for this from the situation that is when I do that I had to divide by excellent both sides. So I'll be getting why why prime By X. This is equal to negative basically it's all for -8. So I'm going to take this next year since I already have an A to A over here. I can simply substitute this negative the value of 98 over here. So when I do that, Albert getting why Y double prime press Y prime all squared this is equal to the value of next to A. Is why why prime boy X. So now we have to get rid of this fraction. So let's multiply both sides by X. And when we do that we'll be getting X. Y. I will prime plus X times why prime square is equal to Y. Y. Print so let's bring every term on one side that is onto the left hand side. So you can rewrite the situation as X. Y. Y double prime plus X times of why prime square is the minus mhm minus Y Y. Prime equal to zero. So you can bring this to the side and when you bring this to the side, this will become negative. So this is basically the required equation why the differential equation for the given family, of course y squared equal to eight times of b squared minus x squared, and this is the answer for this question.

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