Check sibling questions

Ex 9.5, 17 - Which is a homogeneous differential equation

Ex 9.5, 17 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.5, 17 - Chapter 9 Class 12 Differential Equations - Part 3
Ex 9.5, 17 - Chapter 9 Class 12 Differential Equations - Part 4
Ex 9.5, 17 - Chapter 9 Class 12 Differential Equations - Part 5
Ex 9.5, 17 - Chapter 9 Class 12 Differential Equations - Part 6

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Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) (4π‘₯+6𝑦+5)π‘‘π‘¦βˆ’(3𝑦+2π‘₯+4)𝑑π‘₯=0 (B) (π‘₯𝑦)𝑑π‘₯βˆ’(π‘₯^3+𝑦^3 )𝑑𝑦=0 (C) (π‘₯^3+2𝑦^2 )𝑑π‘₯+2π‘₯𝑦 𝑑𝑦=0 (D) 𝑦^2 𝑑π‘₯+(π‘₯^2+π‘₯π‘¦βˆ’π‘¦^2 )𝑑𝑦=0 Let us check each equation one by one Checking (A) Differential equation can be written as (4π‘₯+6𝑦+5)π‘‘π‘¦βˆ’(3𝑦+2π‘₯+4)𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = ((3𝑦 + 2π‘₯ + 4))/((4π‘₯ + 6𝑦 + 5)) Let F(x, y) = 𝑑𝑦/𝑑π‘₯ = ((3𝑦 + 2π‘₯ + 4))/((4π‘₯ + 6𝑦 + 5)) Finding F(πœ†x, πœ†y) F(πœ†x, πœ†y) = (2πœ†π‘₯ + 3πœ†π‘¦ + 4)/(4πœ†π‘₯ + 6πœ†π‘¦ + 5) β‰  πœ†Β° F(x, y) ∴ The given equation is not homogenous Checking (B) (B) Differential equation can be written as (π‘₯𝑦)𝑑π‘₯βˆ’(π‘₯^3+𝑦^3 )𝑑𝑦 = 0 𝑑𝑦/𝑑π‘₯ = π‘₯𝑦/(π‘₯^3 + 𝑦^3 ) Let F(x, y) = 𝑑𝑦/𝑑π‘₯ = π‘₯𝑦/(π‘₯^3 + 𝑦^3 ) Finding F(πœ†x, πœ†y) F(πœ†x, πœ†y) = (πœ†π‘₯ πœ†π‘¦)/(πœ†^3 π‘₯^3 + πœ†^3 𝑦^3 ) = (πœ†^2 π‘₯𝑦)/(πœ†^3 [π‘₯^3 + 𝑦^3 ] ) = π‘₯𝑦/πœ†(π‘₯^3+𝑦^3 ) β‰  πœ†Β° F(x, y) ∴ The given equation is not homogenous Checking (C) (π‘₯^3+2𝑦^2 )𝑑π‘₯+2π‘₯𝑦 𝑑𝑦=0 (x3 + 2y2) dx = βˆ’2xy dy 𝑑𝑦/𝑑π‘₯ = (βˆ’(π‘₯^3 + 2𝑦^2))/2π‘₯𝑦 Let F(x, y) = 𝑑𝑦/𝑑π‘₯ = (βˆ’(π‘₯^3 + 2𝑦^2))/2π‘₯𝑦 Finding F(πœ†x, πœ†y) F(πœ†x, πœ†y) = (βˆ’(πœ†^3 π‘₯^3 + 2πœ†^2 𝑦^2))/2πœ†π‘₯πœ†π‘¦ = (βˆ’γ€–6π‘₯γ€—^3 + 2𝑦^2)/2π‘₯𝑦 β‰  πœ†Β° F(x, y) ∴ The given equation is not homogenous Checking (D) y2 dx + (x2 βˆ’ xy βˆ’ y2) dy = 0 y2 dx = (x2 βˆ’ xy βˆ’ y2)dy 𝑑𝑦/𝑑π‘₯ = 𝑦^2/(π‘₯^2 βˆ’ π‘₯𝑦 βˆ’ 𝑦2) Let F(x, y) = 𝑑𝑦/𝑑π‘₯ = 𝑦^2/(π‘₯^2 βˆ’ π‘₯𝑦 βˆ’ 𝑦2) Finding F(πœ†x, πœ†y) F(πœ†x, πœ†y) = γ€–βˆ’πœ†^(2 ) 𝑦〗^2/(πœ†^(2 ) (π‘₯^2 βˆ’ π‘₯𝑦 βˆ’ 𝑦2)) = 𝑦^2/(π‘₯^2 βˆ’ π‘₯𝑦 βˆ’ 𝑦2) = πœ†Β°F (x, y) F (x, y) is π‘Ž homogenous function of degree zero. ∴ Given equation is a homogenous differential equation. Hence, (D) is the correct answer.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.