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

Transcript

Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) (4๐‘ฅ+6๐‘ฆ+5)๐‘‘๐‘ฆโˆ’(3๐‘ฆ+2๐‘ฅ+4)๐‘‘๐‘ฅ=0 (B) (๐‘ฅ๐‘ฆ)๐‘‘๐‘ฅโˆ’(๐‘ฅ^3+๐‘ฆ^3 )๐‘‘๐‘ฆ=0 (C) (๐‘ฅ^3+2๐‘ฆ^2 )๐‘‘๐‘ฅ+2๐‘ฅ๐‘ฆ ๐‘‘๐‘ฆ=0 (D) ๐‘ฆ^2 ๐‘‘๐‘ฅ+(๐‘ฅ^2+๐‘ฅ๐‘ฆโˆ’๐‘ฆ^2 )๐‘‘๐‘ฆ=0 Let us check each equation one by one Checking (A) Differential equation can be written as (4๐‘ฅ+6๐‘ฆ+5)๐‘‘๐‘ฆโˆ’(3๐‘ฆ+2๐‘ฅ+4)๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ((3๐‘ฆ + 2๐‘ฅ + 4))/((4๐‘ฅ + 6๐‘ฆ + 5)) Let F(x, y) = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ((3๐‘ฆ + 2๐‘ฅ + 4))/((4๐‘ฅ + 6๐‘ฆ + 5)) Finding F(๐œ†x, ๐œ†y) F(๐œ†x, ๐œ†y) = (2๐œ†๐‘ฅ + 3๐œ†๐‘ฆ + 4)/(4๐œ†๐‘ฅ + 6๐œ†๐‘ฆ + 5) โ‰  ๐œ†ยฐ F(x, y) โˆด The given equation is not homogenous Checking (B) (B) Differential equation can be written as (๐‘ฅ๐‘ฆ)๐‘‘๐‘ฅโˆ’(๐‘ฅ^3+๐‘ฆ^3 )๐‘‘๐‘ฆ = 0 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฅ๐‘ฆ/(๐‘ฅ^3 + ๐‘ฆ^3 ) Let F(x, y) = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฅ๐‘ฆ/(๐‘ฅ^3 + ๐‘ฆ^3 ) Finding F(๐œ†x, ๐œ†y) F(๐œ†x, ๐œ†y) = (๐œ†๐‘ฅ ๐œ†๐‘ฆ)/(๐œ†^3 ๐‘ฅ^3 + ๐œ†^3 ๐‘ฆ^3 ) = (๐œ†^2 ๐‘ฅ๐‘ฆ)/(๐œ†^3 [๐‘ฅ^3 + ๐‘ฆ^3 ] ) = ๐‘ฅ๐‘ฆ/๐œ†(๐‘ฅ^3+๐‘ฆ^3 ) โ‰  ๐œ†ยฐ F(x, y) โˆด The given equation is not homogenous Checking (C) (๐‘ฅ^3+2๐‘ฆ^2 )๐‘‘๐‘ฅ+2๐‘ฅ๐‘ฆ ๐‘‘๐‘ฆ=0 (x3 + 2y2) dx = โˆ’2xy dy ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’(๐‘ฅ^3 + 2๐‘ฆ^2))/2๐‘ฅ๐‘ฆ Let F(x, y) = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’(๐‘ฅ^3 + 2๐‘ฆ^2))/2๐‘ฅ๐‘ฆ Finding F(๐œ†x, ๐œ†y) F(๐œ†x, ๐œ†y) = (โˆ’(๐œ†^3 ๐‘ฅ^3 + 2๐œ†^2 ๐‘ฆ^2))/2๐œ†๐‘ฅ๐œ†๐‘ฆ = (โˆ’ใ€–6๐‘ฅใ€—^3 + 2๐‘ฆ^2)/2๐‘ฅ๐‘ฆ โ‰  ๐œ†ยฐ F(x, y) โˆด The given equation is not homogenous Checking (D) y2 dx + (x2 โˆ’ xy โˆ’ y2) dy = 0 y2 dx = (x2 โˆ’ xy โˆ’ y2)dy ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฆ^2/(๐‘ฅ^2 โˆ’ ๐‘ฅ๐‘ฆ โˆ’ ๐‘ฆ2) Let F(x, y) = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฆ^2/(๐‘ฅ^2 โˆ’ ๐‘ฅ๐‘ฆ โˆ’ ๐‘ฆ2) Finding F(๐œ†x, ๐œ†y) F(๐œ†x, ๐œ†y) = ใ€–โˆ’๐œ†^(2 ) ๐‘ฆใ€—^2/(๐œ†^(2 ) (๐‘ฅ^2 โˆ’ ๐‘ฅ๐‘ฆ โˆ’ ๐‘ฆ2)) = ๐‘ฆ^2/(๐‘ฅ^2 โˆ’ ๐‘ฅ๐‘ฆ โˆ’ ๐‘ฆ2) = ๐œ†ยฐF (x, y) F (x, y) is ๐‘Ž homogenous function of degree zero. โˆด Given equation is a homogenous differential equation. Hence, (D) is the correct answer.

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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.