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Last updated at Dec. 11, 2019 by Teachoo

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Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) (4๐ฅ+6๐ฆ+5)๐๐ฆโ(3๐ฆ+2๐ฅ+4)๐๐ฅ=0 (B) (๐ฅ๐ฆ)๐๐ฅโ(๐ฅ^3+๐ฆ^3 )๐๐ฆ=0 (C) (๐ฅ^3+2๐ฆ^2 )๐๐ฅ+2๐ฅ๐ฆ ๐๐ฆ=0 (D) ๐ฆ^2 ๐๐ฅ+(๐ฅ^2+๐ฅ๐ฆโ๐ฆ^2 )๐๐ฆ=0 Let us check each equation one by one Checking (A) Differential equation can be written as (4๐ฅ+6๐ฆ+5)๐๐ฆโ(3๐ฆ+2๐ฅ+4)๐๐ฅ ๐๐ฆ/๐๐ฅ = ((3๐ฆ + 2๐ฅ + 4))/((4๐ฅ + 6๐ฆ + 5)) Let F(x, y) = ๐๐ฆ/๐๐ฅ = ((3๐ฆ + 2๐ฅ + 4))/((4๐ฅ + 6๐ฆ + 5)) Finding F(๐x, ๐y) F(๐x, ๐y) = (2๐๐ฅ + 3๐๐ฆ + 4)/(4๐๐ฅ + 6๐๐ฆ + 5) โ ๐ยฐ F(x, y) โด The given equation is not homogenous Checking (B) (B) Differential equation can be written as (๐ฅ๐ฆ)๐๐ฅโ(๐ฅ^3+๐ฆ^3 )๐๐ฆ = 0 ๐๐ฆ/๐๐ฅ = ๐ฅ๐ฆ/(๐ฅ^3 + ๐ฆ^3 ) Let F(x, y) = ๐๐ฆ/๐๐ฅ = ๐ฅ๐ฆ/(๐ฅ^3 + ๐ฆ^3 ) Finding F(๐x, ๐y) F(๐x, ๐y) = (๐๐ฅ ๐๐ฆ)/(๐^3 ๐ฅ^3 + ๐^3 ๐ฆ^3 ) = (๐^2 ๐ฅ๐ฆ)/(๐^3 [๐ฅ^3 + ๐ฆ^3 ] ) = ๐ฅ๐ฆ/๐(๐ฅ^3+๐ฆ^3 ) โ ๐ยฐ F(x, y) โด The given equation is not homogenous Checking (C) (๐ฅ^3+2๐ฆ^2 )๐๐ฅ+2๐ฅ๐ฆ ๐๐ฆ=0 (x3 + 2y2) dx = โ2xy dy ๐๐ฆ/๐๐ฅ = (โ(๐ฅ^3 + 2๐ฆ^2))/2๐ฅ๐ฆ Let F(x, y) = ๐๐ฆ/๐๐ฅ = (โ(๐ฅ^3 + 2๐ฆ^2))/2๐ฅ๐ฆ Finding F(๐x, ๐y) F(๐x, ๐y) = (โ(๐^3 ๐ฅ^3 + 2๐^2 ๐ฆ^2))/2๐๐ฅ๐๐ฆ = (โใ6๐ฅใ^3 + 2๐ฆ^2)/2๐ฅ๐ฆ โ ๐ยฐ F(x, y) โด The given equation is not homogenous Checking (D) y2 dx + (x2 โ xy โ y2) dy = 0 y2 dx = (x2 โ xy โ y2)dy ๐๐ฆ/๐๐ฅ = ๐ฆ^2/(๐ฅ^2 โ ๐ฅ๐ฆ โ ๐ฆ2) Let F(x, y) = ๐๐ฆ/๐๐ฅ = ๐ฆ^2/(๐ฅ^2 โ ๐ฅ๐ฆ โ ๐ฆ2) Finding F(๐x, ๐y) F(๐x, ๐y) = ใโ๐^(2 ) ๐ฆใ^2/(๐^(2 ) (๐ฅ^2 โ ๐ฅ๐ฆ โ ๐ฆ2)) = ๐ฆ^2/(๐ฅ^2 โ ๐ฅ๐ฆ โ ๐ฆ2) = ๐ยฐF (x, y) F (x, y) is ๐ homogenous function of degree zero. โด Given equation is a homogenous differential equation. Hence, (D) is the correct answer.

Chapter 9 Class 12 Differential Equations

Serial order wise

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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.