Example 21 - Solve (x dy - ydx) y sin (y/x) = (ydx + xdy) (2025)

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1. Chapter 9 Class 12 Differential Equations
2. Serial order wise
3. Examples

Examples

Example 1 (i)

Example 1 (ii) Important

Example 1 (iii) Important

Example 2

Example 3 Important

Example 4

Example 5

Example 6

Example 7 Important

Example 8

Example 9 Important

Example 10 Important

Example 11

Example 12 Important

Example 13 Important

Example 14

Example 15 Important

Example 16

Example 17 Important

Example 18 Important

Example 19

Example 20

Example 21 Important You are here

Example 22 Important

Question 1

Question 2

Question 3 Important

Question 4

Question 5

Question 6

Miscellaneous→

Chapter 9 Class 12 Differential Equations

Serial order wise

• Ex 9.1
• Ex 9.2
• Ex 9.3
• Ex 9.4
• Ex 9.5

• Examples

• Miscellaneous
• Case Based Questions (MCQ)
• Forming Differential equations

Last updated at Dec. 16, 2024 by Teachoo

Next: Example 22 Important → Go Ad-free

Transcript

Example 21 Solve the differential equation (𝑥 𝑑𝑦−𝑦 𝑑𝑥)𝑦 𝑠𝑖𝑛(𝑦/𝑥)=(𝑦 𝑑𝑥+𝑥 𝑑𝑦)𝑥 cos⁡(𝑦/𝑥) (𝑥 𝑑𝑦−𝑦 𝑑𝑥)𝑦 𝑠𝑖𝑛(𝑦/𝑥)=(𝑦 𝑑𝑥+𝑥 𝑑𝑦)𝑥 cos⁡(𝑦/𝑥) 𝑥 𝑦 sin⁡〖(𝑦/𝑥)𝑑𝑦−𝑦^2 𝑠𝑖𝑛(𝑦/𝑥) 〗 𝑑𝑥=𝑥𝑦 cos⁡〖(𝑦/𝑥)𝑑𝑥+𝑥^2 〗 cos⁡(𝑦/𝑥)𝑑𝑦[𝑥 𝑦 sin⁡〖(𝑦/𝑥)−𝑥^2 𝑐𝑜𝑠(𝑦/𝑥) 〗 ]𝑑𝑦=[𝑥𝑦 cos⁡〖(𝑦/𝑥)+𝑦^2 〗 sin⁡(𝑦/𝑥) ]𝑑𝑥𝒅𝒚/𝒅𝒙 = (𝒙𝒚 𝐜𝐨𝐬⁡〖(𝒚/𝒙) + 𝒚^𝟐 𝐬𝐢𝐧⁡(𝒚/𝒙) 〗)/(𝒙𝒚 𝐬𝐢𝐧⁡〖(𝒚/𝒙) − 𝒙^𝟐 𝒄𝒐𝒔(𝒚/𝒙)〗 )Dividing numerator & denominator by x2𝑑𝑦/𝑑𝑥 = (𝑦/𝑥 cos⁡〖(𝑦/𝑥) + (𝑦/𝑥)^2 cos⁡(𝑦/𝑥) 〗)/(𝑦/𝑥 sin⁡〖(𝑦/𝑥) − cos⁡(𝑦/𝑥) 〗 )Putting y = vx.Differentiating w.r.t. x𝒅𝒚/𝒅𝒙 = 𝒙 𝒅𝒗/𝒅𝒙 + vPutting value of 𝑑𝑦/𝑑𝑥 and y in (1)𝑑𝑦/𝑑𝑥 = (𝑦/𝑥 cos⁡〖(𝑦/𝑥) + (𝑦/𝑥)^2 cos⁡(𝑦/𝑥) 〗)/(𝑦/𝑥 sin⁡〖(𝑦/𝑥) − cos⁡(𝑦/𝑥) 〗 )v + (𝑥 𝑑𝑣)/𝑑𝑥=(𝑣𝑥/𝑥 cos⁡〖(𝑣𝑥/𝑥) + (𝑣^2 𝑥^2)/𝑥^2 sin⁡(𝑣𝑥/𝑥) 〗)/(𝑣𝑥/𝑥 sin⁡〖(𝑣𝑥/𝑥) − 𝑐𝑜𝑠(𝑣𝑥/𝑥)〗 )v + (𝒙 𝒅𝒗)/𝒅𝒙 = (𝒗 𝐜𝐨𝐬⁡〖𝒗 + 𝒗^𝟐 𝐬𝐢𝐧⁡𝒗 〗)/(𝒗 𝐬𝐢𝐧⁡〖𝒗 − 𝐜𝐨𝐬⁡𝒗 〗 ) x ( 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡𝑣 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) − v x ( 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡〖𝑣 − 𝑣(𝑣 sin⁡〖𝑣 − cos⁡〖𝑣)〗 〗 〗 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 ) x ( 𝑑𝑣)/𝑑𝑥 = (𝑣 cos⁡〖𝑣 + 𝑣^2 sin⁡〖𝑣 − 𝑣^2 sin 𝑣〗 + 𝑣 cos⁡𝑣 〗)/(𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗 )( 𝒙 𝒅𝒗)/𝒅𝒙 = (𝟐𝒗 𝒄𝒐𝒔⁡𝒗)/(𝒗 𝒔𝒊𝒏⁡〖𝒗 − 𝒄𝒐𝒔⁡𝒗 〗 ) ((𝑣 sin⁡〖𝑣 − cos⁡𝑣 〗)/〖v cos〗⁡𝑣 )𝑑𝑣=2 𝑑𝑥/𝑥((𝑣 sin⁡𝑣)/〖v cos〗⁡𝑣 −cos⁡𝑣/〖v cos〗⁡𝑣 ) dv = 2 𝑑𝑥/𝑥(𝒕𝒂𝒏⁡𝒗−𝟏/𝒗) dv = 2 𝒅𝒙/𝒙Integrating both sides∫1▒〖(tan⁡〖𝑣 −1/𝑣〗 )𝑑𝑣=2∫1▒𝑑𝑥/𝑥〗 ∫1▒tan⁡〖𝑣 𝑑𝑣 − 〗 ∫1▒𝑑𝑣/𝑣 = 2 ∫1▒𝑑𝑥/𝑥log |𝒔𝒆𝒄⁡𝒗 |−𝐥𝐨𝐠⁡〖|𝒗|〗 = 2 log |𝒙| + log 𝐂log |sec⁡𝑣 |−log⁡〖|𝑣|〗 = log |𝑥^2 | + log Clog |𝒔𝒆𝒄⁡𝒗/𝒗| = log 𝒙^𝟐 + log 𝐂log |sec⁡𝑣/𝑣| = log 𝑥^2 𝑐𝒔𝒆𝒄⁡𝒗/𝒗 = 𝒙^𝟐 𝒄Putting back value of v = 𝑦/𝑥〖sec 〗⁡(𝑦/𝑥)/((𝑦/𝑥) ) = 𝑥^2 𝑐sec⁡(𝑦/𝑥) = (𝑦/𝑥) 𝑥^2 𝑐sec (𝒚/𝒙) = C xy

1. Chapter 9 Class 12 Differential Equations
2. Serial order wise

Examples

Example 1 (i)



Example 1 (ii) Important



Example 1 (iii) Important



Example 2



Example 3 Important



Example 4



Example 5



Example 6



Example 7 Important



Example 8



Example 9 Important



Example 10 Important



Example 11



Example 12 Important



Example 13 Important



Example 14



Example 15 Important



Example 16



Example 17 Important



Example 18 Important



Example 19



Example 20



Example 21 Important You are here



Example 22 Important



Question 1



Question 2



Question 3 Important



Question 4



Question 5



Question 6



Miscellaneous→

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo