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


Transcript

Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) (4𝑥+6𝑦+5)𝑑𝑦−(3𝑦+2𝑥+4)𝑑𝑥=0 (B) (𝑥𝑦)𝑑𝑥−(𝑥^3+𝑦^3 )𝑑𝑦=0 (C) (𝑥^3+2𝑦^2 )𝑑𝑥+2𝑥𝑦 𝑑𝑦=0 (D) 𝑦^2 𝑑𝑥+(𝑥^2+𝑥𝑦−𝑦^2 )𝑑𝑦=0 Let us check each equation one by one Checking (A) Differential equation can be written as (4𝑥+6𝑦+5)𝑑𝑦−(3𝑦+2𝑥+4)𝑑𝑥 𝑑𝑦/𝑑𝑥 = ((3𝑦 + 2𝑥 + 4))/((4𝑥 + 6𝑦 + 5)) Let F(x, y) = 𝑑𝑦/𝑑𝑥 = ((3𝑦 + 2𝑥 + 4))/((4𝑥 + 6𝑦 + 5)) Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = (2𝜆𝑥 + 3𝜆𝑦 + 4)/(4𝜆𝑥 + 6𝜆𝑦 + 5) ≠ 𝜆° F(x, y) ∴ The given equation is not homogenous Checking (B) (B) Differential equation can be written as (𝑥𝑦)𝑑𝑥−(𝑥^3+𝑦^3 )𝑑𝑦 = 0 𝑑𝑦/𝑑𝑥 = 𝑥𝑦/(𝑥^3 + 𝑦^3 ) Let F(x, y) = 𝑑𝑦/𝑑𝑥 = 𝑥𝑦/(𝑥^3 + 𝑦^3 ) Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = (𝜆𝑥 𝜆𝑦)/(𝜆^3 𝑥^3 + 𝜆^3 𝑦^3 ) = (𝜆^2 𝑥𝑦)/(𝜆^3 [𝑥^3 + 𝑦^3 ] ) = 𝑥𝑦/𝜆(𝑥^3+𝑦^3 ) ≠ 𝜆° F(x, y) ∴ The given equation is not homogenous Checking (C) (𝑥^3+2𝑦^2 )𝑑𝑥+2𝑥𝑦 𝑑𝑦=0 (x3 + 2y2) dx = −2xy dy 𝑑𝑦/𝑑𝑥 = (−(𝑥^3 + 2𝑦^2))/2𝑥𝑦 Let F(x, y) = 𝑑𝑦/𝑑𝑥 = (−(𝑥^3 + 2𝑦^2))/2𝑥𝑦 Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = (−(𝜆^3 𝑥^3 + 2𝜆^2 𝑦^2))/2𝜆𝑥𝜆𝑦 = (−〖6𝑥〗^3 + 2𝑦^2)/2𝑥𝑦 ≠ 𝜆° F(x, y) ∴ The given equation is not homogenous Checking (D) y2 dx + (x2 − xy − y2) dy = 0 y2 dx = (x2 − xy − y2)dy 𝑑𝑦/𝑑𝑥 = 𝑦^2/(𝑥^2 − 𝑥𝑦 − 𝑦2) Let F(x, y) = 𝑑𝑦/𝑑𝑥 = 𝑦^2/(𝑥^2 − 𝑥𝑦 − 𝑦2) Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = 〖−𝜆^(2 ) 𝑦〗^2/(𝜆^(2 ) (𝑥^2 − 𝑥𝑦 − 𝑦2)) = 𝑦^2/(𝑥^2 − 𝑥𝑦 − 𝑦2) = 𝜆°F (x, y) F (x, y) is 𝑎 homogenous function of degree zero. ∴ Given equation is a homogenous differential equation. Hence, (D) is the correct answer.

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Davneet Singh

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.