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Question 21 (OR 2
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nd
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question)
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Find the general solution of the differential equation:

dx/dy = (y tany - x tany - xy) / (y tany)

Last updated at Oct. 1, 2019 by Teachoo

**
Question 21 (OR 2
**
**
nd
**
**
question)
**

Find the general solution of the differential equation:

dx/dy = (y tany - x tany - xy) / (y tany)

Transcript

Question 21 (OR 2nd question) Find the general solution of the differential equation: 𝑑𝑥/𝑑𝑦=(𝑦 tan𝑦 − 𝑥 tan𝑦 − 𝑥𝑦)/(𝑦 tan𝑦 ) 𝑑𝑥/𝑑𝑦=(𝑦 tan𝑦 − 𝑥 tan𝑦 − 𝑥𝑦)/(𝑦 tan𝑦 ) 𝑑𝑥/𝑑𝑦=(𝑦 tan𝑦)/(𝑦 tan𝑦 )−(𝑥 tan𝑦 )/(𝑦 tan𝑦 )−𝑥𝑦/(𝑦 tan𝑦 ) 𝑑𝑥/𝑑𝑦=1−𝑥/𝑦−𝑥/tan𝑦 𝑑𝑥/𝑑𝑦+𝑥/𝑦+𝑥/tan𝑦 =1 𝑑𝑥/𝑑𝑦+𝑥(1/𝑦+1/tan𝑦 )=1 Differential equation is of the form 𝑑𝑥/𝑑𝑦 + P1 x = Q1 where P1 = 1/𝑦+1/tan𝑦 & Q1 = 1 Now, IF = 𝑒^∫1▒〖𝑝_1 𝑑𝑦〗 IF = e^∫1▒〖(1/𝑦 + 1/tan𝑦 )𝑑𝑦" " 〗 IF = e^(∫1▒〖1/𝑦 𝑑𝑦〗 +∫1▒〖1/tan𝑦 𝑑𝑦〗) IF = e^(∫1▒〖1/𝑦 𝑑𝑦〗 +∫1▒〖cot𝑦 𝑑𝑦〗) IF = e^(log𝑦 + logsin𝑦 ) IF = e^〖log 〗〖(𝑦 sin𝑦)〗 IF = y sin y Solution is x(IF) = ∫1▒〖(𝑄×𝐼𝐹)𝑑𝑦+𝐶 〗 x (y sin y) =∫1▒〖1×𝑦 sin𝑦 〗 𝑑𝑦+𝐶 xy sin y =∫1▒〖𝑦 sin𝑦 〗 𝑑𝑦+𝐶 (As ∫1▒cot𝑥 𝑑𝑥=logsin𝑥 ) (As log a + log b = log ab) We know that ∫1▒〖𝑓(𝑦) 𝑔(𝑦) 〗 𝑑𝑦=𝑓(𝑦) ∫1▒𝑔(𝑦) 𝑑𝑦−∫1▒(𝑓^′ 𝑦∫1▒𝑔(𝑦) 𝑑𝑦) 𝑑𝑦 Putting f(y) = y and g(y) = sin y xy sin y =𝑦" " ∫1▒sin𝑦 𝑑𝑦−∫1▒(𝑑(𝑦)/𝑑𝑦 ∫1▒〖sin𝑦 𝑑𝑦〗) 𝑑𝑦 xy sin y =−𝑦 cos𝑦 − ∫1▒〖−cos𝑦 𝑑𝑦〗 xy sin y =−𝑦 cos𝑦+ ∫1▒〖cos𝑦 𝑑𝑦〗 xy sin y =−𝑦 cos𝑦+sin𝑦+𝐶 xy sin y =sin𝑦−𝑦 cos𝑦+𝐶 x = (𝒔𝒊𝒏𝒚 − 𝒚 𝒄𝒐𝒔𝒚 + 𝑪" " )/(𝒚 𝐬𝐢𝐧𝒚 )

CBSE Class 12 Sample Paper for 2019 Boards

Paper Summary

Question 1

Question 2

Question 3

Question 4 (Or 1st)

Question 4 (Or 2nd)

Question 5

Question 6

Question 7

Question 8 (Or 1st)

Question 8 (Or 2nd)

Question 9

Question 10 (Or 1st)

Question 10 (Or 2nd)

Question 11

Question 12 (Or 1st)

Question 12 (Or 2nd)

Question 13 (Or 1st)

Question 13 (Or 2nd)

Question 14

Question 15

Question 16 (Or 1st)

Question 16 (Or 2nd)

Question 17

Question 18

Question 19

Question 20

Question 21 (Or 1st)

Question 21 (Or 2nd) You are here

Question 22

Question 23

Question 24 (Or 1st)

Question 24 (Or 2nd)

Question 25

Question 26 (Or 1st)

Question 26 (Or 2nd)

Question 27 (Or 1st)

Question 27 (Or 2nd)

Question 28

Question 29

Class 12

Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.