Problem

For each of the differential equations in Exercis…

04:35

Problem 4 Easy Difficulty

For each of the differential equations in Exercises 1 to 10 , find the general solution:
$\sec ^{2} x \tan y d x+\sec ^{2} y \tan x d y=0$

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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Problem 1

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Video Transcript

you will not find the general solution of this given differential equation. That is second square X. Stand where the X plus seconds squared Y to annex the way equal to zero using the variable separable minted. So in this method there are about three steps that we could follow. The first step is basically we just have to simplify the terms on the left side and arches using algebraic or geometric identities. And in the second step, which is the most important step of this method, we separate all white terms to one side of the equation and all X terms to the other side of the equation. And finally we integrate both sides with respect to X and preferably we solve forward in case if it is solvable for way otherwise we retain the solution equation in terms of X and Y. So let's go ahead and apply those steps one by one. Notice that we don't have any terms where we could apply algebraic or techno metric identities and simplified. So we skipped the first step for this particular problem and move on to the next step. The next step, What I'm going to do is I'm going to retain this term as it is and transfer this term to the other side of the equation. So when I do that, I'll be getting a second squared y an ex do Y. This will be the negative of this term. That is a negative of second squared x. Dan. Why? Dx observe that? We still haven't separated the variable successfully because if you look this side, we have a combination of why as well as that as opposed to external that is we have seconds quite why, which is a white term and dynamics, which is an extreme. And as well as the way the same thing we have it on the right side which is a combination of X terms and white terms. So what we do now is we divide both sides by this term that is considered this uh tan Y. So we're going to divide boxes of by this term and when we do that we'll be getting second squared y over then why? And similarly we are going to divide ah both sides by this time. So we'll do this in just one step and when we do that this term will get cancelled over here since we do it by the term and this side will be getting minus off second squared X. Since we debate by this term as well, it will be dan X and this term will get canceled. This term will get canceled since we divide both sides by tan Y. So I can write on this like this and if you observe now we have successfully separated the variables as you can see on the left side, we have all terms in why as plus on the origins, we have all the terms in terms of. Excellently. So this is um we help separated the variables, we can go ahead and do the third step that is, we have to integrate it. So let me write the situation that is we have seconds quite why over dan way the Y. Is equal to negative of uh second squared x over 10 X. New york's with an integrate both sides like this. And so this employees when you integrate this side that is on the website we have second square Y over tan Y Dy So integration of this will be natural algorithm of absolute value of tan Y. And this site we already have a negative so I put the negative and similarly we have this integration of seconds cortex over 10 X. So this will be natural logo tomorrow absolute value of 10 X. And we will have we will get integrating constancy. So usually we will put the constant asi but this time I'm going to put the constant as natural algorithm of constant absolute value of seat so for some purpose so that we could simplify this equation. Now I add this term to both sides. That is uh this term make you do of uh land natural algorithm of tenants to this to the both sides and when I do that I'll be getting natural algorithm off dan way, absolute value of tramway plus natural log on tomorrow 10 X is equal to natural rock bottom off, absolute value of C. So here we could uh play the properties of natural uh algorithm that is a natural rhythm of E press natural logarithms be this could be simplified this natural rock bottom of A B. So when we apply this property of natural Baltimore here, this will become natural, good time off. Turn white and 10 x. I will write down discuss panics. And done. We absolutely of Conway and this side we have naturally longer them off absolute value of C. We can now go ahead and remove the natural algorithm or both sides because we have a natural rhythm over here as well as natural rhythm over here. So you can simply remove this natural rhythm. So when we do that we'll get to annex dan White is equal to see. So this is the general solution of the given differential equation.

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