For what value of n is the following a homogeneous differential equation: dy/dx = x ^{ 3 } - yn / x ^{ 2 } y + xy ^{ 2 }
Last updated at Oct. 26, 2020 by Teachoo
Transcript
Question 9 (Choice 2) For what value of n is the following a homogeneous differential equation: ๐๐ฆ/๐๐ฅ = (๐ฅ^3 โ ๐ฆ^๐)/(๐ฅ^2 ๐ฆ + ๐ฅ๐ฆ^2 ) Given ๐๐ฆ/๐๐ฅ = (๐ฅ^3 โ ๐ฆ^๐)/(๐ฅ^2 ๐ฆ + ๐ฅ๐ฆ^2 ) Thus, F(x, y) = (๐ฅ^3 โ ๐ฆ^๐)/(๐ฅ^2 ๐ฆ + ๐ฅ๐ฆ^2 ) Since given equation is homegenous F(๐x, ๐y) = ๐^0 F(x, y) (ใ(๐๐ฅ)ใ^3 โใ (๐๐ฆ)ใ^๐)/((๐๐ฅ)^2 (๐๐ฆ) +(๐๐ฅ)ใ(๐๐ฆ)ใ^2 )=(๐ฅ^3 โ ๐ฆ^๐)/(๐ฅ^2 ๐ฆ + ๐ฅ๐ฆ^2 ) (๐^3 ๐ฅ^3 โ ใ(๐๐ฆ)ใ^๐)/(๐^3 ๐ฅ^2 ๐ฆ +๐^3 ๐ฅ๐ฆ^2 )=(๐ฅ^3 โ ๐ฆ^๐)/(๐ฅ^2 ๐ฆ + ๐ฅ๐ฆ^2 ) (๐^๐ ๐^๐ โ ใ(๐๐)ใ^๐)/(๐^๐ (๐^๐ ๐ + ๐๐^๐))=(๐^๐ โ ๐^๐)/(๐^๐ ๐ + ๐๐^๐ ) This is only possible when n = 3
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CBSE Class 12 Sample Paper for 2021 Boards
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