For what value of n is the following a homogeneous differential equation: dy/dx = x 3 - yn / x 2 y + xy 2

For what value of n is following a homogeneous differential equation

Question 9  (Choice 2) - CBSE Class 12 Sample Paper for 2021 Boards - Part 2

 

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

Transcript

Question 9 (Choice 2) For what value of n is the following a homogeneous differential equation: ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘ฅ^3 โˆ’ ๐‘ฆ^๐‘›)/(๐‘ฅ^2 ๐‘ฆ + ๐‘ฅ๐‘ฆ^2 ) Given ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘ฅ^3 โˆ’ ๐‘ฆ^๐‘›)/(๐‘ฅ^2 ๐‘ฆ + ๐‘ฅ๐‘ฆ^2 ) Thus, F(x, y) = (๐‘ฅ^3 โˆ’ ๐‘ฆ^๐‘›)/(๐‘ฅ^2 ๐‘ฆ + ๐‘ฅ๐‘ฆ^2 ) Since given equation is homegenous F(๐œ†x, ๐œ†y) = ๐œ†^0 F(x, y) (ใ€–(๐œ†๐‘ฅ)ใ€—^3 โˆ’ใ€– (๐œ†๐‘ฆ)ใ€—^๐‘›)/((๐œ†๐‘ฅ)^2 (๐œ†๐‘ฆ) +(๐œ†๐‘ฅ)ใ€–(๐œ†๐‘ฆ)ใ€—^2 )=(๐‘ฅ^3 โˆ’ ๐‘ฆ^๐‘›)/(๐‘ฅ^2 ๐‘ฆ + ๐‘ฅ๐‘ฆ^2 ) (๐œ†^3 ๐‘ฅ^3 โˆ’ ใ€–(๐œ†๐‘ฆ)ใ€—^๐‘›)/(๐œ†^3 ๐‘ฅ^2 ๐‘ฆ +๐œ†^3 ๐‘ฅ๐‘ฆ^2 )=(๐‘ฅ^3 โˆ’ ๐‘ฆ^๐‘›)/(๐‘ฅ^2 ๐‘ฆ + ๐‘ฅ๐‘ฆ^2 ) (๐€^๐Ÿ‘ ๐’™^๐Ÿ‘ โˆ’ ใ€–(๐€๐’š)ใ€—^๐’)/(๐€^๐Ÿ‘ (๐’™^๐Ÿ ๐’š + ๐’™๐’š^๐Ÿ))=(๐’™^๐Ÿ‘ โˆ’ ๐’š^๐’)/(๐’™^๐Ÿ ๐’š + ๐’™๐’š^๐Ÿ ) This is only possible when n = 3

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

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.