##
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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About the Author

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.