Problem

For each of the differential equations in Exercis…

07:44

Problem 8 Easy Difficulty

For each of the differential equations in Exercises 1 to 10 , find the general solution:
$x^{5} \frac{d y}{d x}=-y^{5}$

Answer

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Calculus 2 / BC

NCERT Class 12 Part 2 - Math

Chapter 9

Differential Equations

Section 4

Formation of a Differential Equation whose General Solution is given

Discussion

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Watch More Solved Questions in Chapter 9

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Video Transcript

we will not find the general solution of this given differential equation that is export five times of the verbal dx is equal to negative oil raised to the power of fight, we're going to do this using this variable separable method in this method we separate the variable, suppose that is we separate all the white terms to one side and all the extremes to the other side. We then integrate with respect to X. And finally we solve for white. So let's go ahead and do this. So the first time we have to separate the variables. So to do that, I'm going to divide both sides of the equation by export five times of my perfect. So when I do this, This side, I'll be getting export five Divided by a country. I don't like this way for five. We have the way by dX. So this side -4, 5 divided by export fi my poor fight. So we can now cancel the few times that is on to the website. You can cancel this expert fight and we can also cancel this way for fight. And so we'll be getting do you want a blank y por file? Times of dx is equal to -1 by export fighting. Uh, we will now multiply both sides by dX because this is the only term that is onto the left side. Other terms are items. So we'll do that. We'll multiply by dX. So that's in place. The sign will be having dy over My Part five is a call to the side will be having dX over explore if I and we have a NATO next in front of it. So now we have successfully separated the variables. If you look at this website we have the way by Aiport boy. And on the right side we have dx by export file which is uh everything in terms of X. So let's go ahead and integrated. Let me write the situation dy over or a poor five is equal to minus of dx over export fight. We integrate with the integration symbol and so we can apply the power rule here. This in fact we can write down as why raised to the power of minus y do y integration. This is a call to minus integration of X rays to the power of -5 DX. So now we can apply the power rule to integrate both says that is the general power rule of integrationists. If you have export and the extent of fear to integrate this, this is equal to x rays. To the power of n plus one Divided by and plus one and and should not be called to minus one. I'm sorry. So and should not be called a -1. So that's the only restriction. So let's do that. We're going to apply this formula. So that food this side when we integrate the website will be getting y raised to the power of negative fight Plus one divided by -5 plus one. Let's say minus off x rays to the power of minus five plus one Started by -5-plus 1. And then we put this integration constant that tasters see. And so Let's keep simplifying this one. So this will be why raised to the power of -4 divided by -4. You c call to this time we already have a negative. This is x rays. To the power of -4 Should be -4 divided by my last book plus C. Okay so we can cancel this negative negative so therefore this is equal to X raised to the power of -40. Before we know multiply both sides of the situation by negative for so that we can get rid of this denominator so therefore we're beginning. Why raised to the power of -4. This is equal to minus four. James off. I'm sorry Since we multiply by -4 this will be minus off be having Minour of x rays to the power of -4 -4 times of C. That's what we'll be having. Yes. Ah verify that because decide to help Explore -4 divided by four and when I multiply this with the -4 I'll be getting -1. Yeah this is correct. So now we can substitute this as a simple constant. We will replace this -4. C replace this -4 C. As big C. This is another constant. So therefore and then we add this negative x negative x rays to the power of four to both sides. So when we do that, we'll be getting x rays to the power of minus book. Plus Why raised to the power of -4 is equal to see. So this is to solution to the given or differential equation.

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