Slide15.JPG

Slide16.JPG
Slide17.JPG Slide18.JPG Slide19.JPG

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

Transcript

Ex 9.5, 15 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : 2๐‘ฅ๐‘ฆ+๐‘ฆ^2โˆ’2๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=0;๐‘ฆ=2 When ๐‘ฅ=1 Differential equation can be written ๐‘Žs 2๐‘ฅ๐‘ฆ+๐‘ฆ^2โˆ’2๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=0 2๐‘ฅ๐‘ฆ+๐‘ฆ^2= 2๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ 2๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=2๐‘ฅ๐‘ฆ+๐‘ฆ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= (2๐‘ฅ๐‘ฆ + ๐‘ฆ^2)/(2๐‘ฅ^2 ) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= ๐‘ฆ/๐‘ฅ + ๐‘ฆ^2/(2๐‘ฅ^2 ) โ€ฆ(1) Let F(x, y) = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฆ/๐‘ฅ + ๐‘ฆ^2/(2๐‘ฅ^2 ) Finding F(๐œ†x, ๐œ†y) F(๐œ†x, ๐œ†y) = ๐œ†๐‘ฆ/๐œ†๐‘ฅ + ใ€–(๐œ†๐‘ฆ)ใ€—^2/(2ใ€–(๐œ†๐‘ฅ)ใ€—^(2 ) ) = ๐‘ฆ/๐‘ฅ + ๐‘ฆ^2/(2๐‘ฅ^2 ) = ๐œ†ยฐ F(x, y) โˆด F(x, y) is a homogenous function of degree zero Putting y = vx Diff w.r.t. x ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = x ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ + v Putting value of ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ and y = vx in (1) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= ๐‘ฆ/๐‘ฅ + ๐‘ฆ^2/(2๐‘ฅ^2 ) ๐‘ฃ+๐‘ฅ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ๐‘ฅ/๐‘ฅ + 1/2 (๐‘ฃ^2 ๐‘ฅ^2)/๐‘ฅ^2 ๐‘ฃ+๐‘ฅ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ+ ๐‘ฃ^2/2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= ๐‘ฆ/๐‘ฅ + ๐‘ฆ^2/(2๐‘ฅ^2 ) ๐‘ฃ+๐‘ฅ ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ+ ๐‘ฃ^2/2 ๐‘ฅ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ^2/2 2๐‘‘๐‘ฃ/๐‘ฃ^2 = ๐‘‘๐‘ฅ/๐‘ฅ Integrating both sides 2โˆซ1โ–’๐‘‘๐‘ฃ/๐‘ฃ^2 "=" โˆซ1โ–’๐‘‘๐‘ฅ/๐‘ฅ 2โˆซ1โ–’ใ€–๐‘ฃ^(โˆ’2) ๐‘‘๐‘ฃ=logโก|๐‘ฅ|+๐‘ใ€— 2 (๐‘ฃ^(โˆ’2 + 1) )/(โˆ’2 + 1) =logโก|๐‘ฅ|+๐‘ 2 (ใ€–๐‘ฃ ใ€—^(โˆ’1) )/(โˆ’1) =logโก|๐‘ฅ|+๐‘ (โˆ’2 )/๐‘ฃ =logโก|๐‘ฅ|+๐‘ 2 (ใ€–๐‘ฃ ใ€—^(โˆ’1) )/(โˆ’1) =logโก|๐‘ฅ|+๐‘ (โˆ’2 )/๐‘ฃ =logโก|๐‘ฅ|+๐‘ Putting value of v = (๐‘ฆ )/๐‘ฅ (โˆ’2๐‘ฅ)/๐‘ฆ = log |๐‘ฅ| + C Now, Putting x = 1 & y = 2 in (2) (โˆ’2(1))/2 = log |1| + C โˆ’1 = 0 + C C = โˆ’1 โ€ฆ(2) (As log 1 = 0) Putting value of C in (2) (โˆ’2๐‘ฅ)/๐‘ฆ = log |๐‘ฅ| + C (โˆ’2๐‘ฅ)/๐‘ฆ = log |๐‘ฅ| โˆ’ 1 y = (โˆ’2๐‘ฅ)/(ใ€–log ใ€—โก|๐‘ฅ| โˆ’ 1) y = ๐Ÿ๐’™/ใ€–๐Ÿ โˆ’ ๐ฅ๐จ๐  ใ€—โก|๐’™|" "

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
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.