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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

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Question 9 (Choice 2) For what value of n is the following a homogeneous differential equation: 𝑑𝑦/𝑑π‘₯ = (π‘₯^3 βˆ’ 𝑦^𝑛)/(π‘₯^2 𝑦 + π‘₯𝑦^2 ) Given 𝑑𝑦/𝑑π‘₯ = (π‘₯^3 βˆ’ 𝑦^𝑛)/(π‘₯^2 𝑦 + π‘₯𝑦^2 ) Thus, F(x, y) = (π‘₯^3 βˆ’ 𝑦^𝑛)/(π‘₯^2 𝑦 + π‘₯𝑦^2 ) Since given equation is homegenous F(πœ†x, πœ†y) = πœ†^0 F(x, y) (γ€–(πœ†π‘₯)γ€—^3 βˆ’γ€– (πœ†π‘¦)γ€—^𝑛)/((πœ†π‘₯)^2 (πœ†π‘¦) +(πœ†π‘₯)γ€–(πœ†π‘¦)γ€—^2 )=(π‘₯^3 βˆ’ 𝑦^𝑛)/(π‘₯^2 𝑦 + π‘₯𝑦^2 ) (πœ†^3 π‘₯^3 βˆ’ γ€–(πœ†π‘¦)γ€—^𝑛)/(πœ†^3 π‘₯^2 𝑦 +πœ†^3 π‘₯𝑦^2 )=(π‘₯^3 βˆ’ 𝑦^𝑛)/(π‘₯^2 𝑦 + π‘₯𝑦^2 ) (𝝀^πŸ‘ 𝒙^πŸ‘ βˆ’ γ€–(π€π’š)γ€—^𝒏)/(𝝀^πŸ‘ (𝒙^𝟐 π’š + π’™π’š^𝟐))=(𝒙^πŸ‘ βˆ’ π’š^𝒏)/(𝒙^𝟐 π’š + π’™π’š^𝟐 ) This is only possible when n = 3

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.