# Ex 9.5, 17 - Chapter 9 Class 12 Differential Equations

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (A) 4𝑥+6𝑦+5𝑑𝑦− 3𝑦+2𝑥+4𝑑𝑥=0 • Differential equation can be written as 4𝑥+6𝑦+5𝑑𝑦− 3𝑦+2𝑥+4𝑑𝑥 𝑑𝑦𝑑𝑥 = (3𝑦 + 2𝑥 + 4)(4𝑥 + 6𝑦 + 5) Let F(x, y) = 𝑑𝑦𝑑𝑥 = (3𝑦 + 2𝑥 + 4)(4𝑥 + 6𝑦 + 5) Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = = 2𝜆𝑥 + 3𝜆𝑦 + 44𝜆𝑥 + 6𝜆𝑦 + 5 ≠ 𝜆° F(x, y) ∴ The given equation is not homogenous Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (B) 𝑥𝑦𝑑𝑥− 𝑥3+ 𝑦3𝑑𝑦=0 (B) Differential equation can be written as 𝑥𝑦𝑑𝑥− 𝑥3+ 𝑦3𝑑𝑦 = 0 𝑑𝑦𝑑𝑥 = 𝑥𝑦 𝑥3 + 𝑦3 Let F(x, y) = 𝑑𝑦𝑑𝑥 = 𝑥𝑦 𝑥3 + 𝑦3 Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = 𝜆𝑥 𝜆𝑦 𝜆3 𝑥3 + 𝜆3 𝑦3 = 𝜆2𝑥𝑦 𝜆3 𝑥3 + 𝑦3 = 𝑥𝑦𝜆 𝑥3+ 𝑦3 ≠ 𝜆° F(x, y) ∴ The given equation is not homogenous Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (C) 𝑥3+2 𝑦2𝑑𝑥+2𝑥𝑦 𝑑𝑦=0 𝑥3+2 𝑦2𝑑𝑥+2𝑥𝑦 𝑑𝑦=0 (x3 + 2y2) dx = −2xy dy 𝑑𝑦𝑑𝑥 = −( 𝑥3+2 𝑦2)2𝑥𝑦 Let F(x, y) = 𝑑𝑦𝑑𝑥 = −( 𝑥3+2 𝑦2)2𝑥𝑦 Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = −( 𝜆3 𝑥3 + 2 𝜆2 𝑦2)2𝜆𝑥𝜆𝑦 = − 6𝑥3 + 2 𝑦22𝑥𝑦 ≠ 𝜆° F(x, y) ∴ The given equation is not homogenous Ex 9.5, 17 Which of the following is a homogeneous differential equation ? (D) 𝑦2𝑑𝑥+ 𝑥2+𝑥𝑦− 𝑦2𝑑𝑦=0 y2 dx + (x2 − xy − y2) dy = 0 y2 dx = (x2 − xy − y2)dy 𝑑𝑦𝑑𝑥 = 𝑦2 𝑥2 − 𝑥𝑦 − 𝑦2 Let F(x, y) = 𝑑𝑦𝑑𝑥 = 𝑦2 𝑥2 − 𝑥𝑦 − 𝑦2 Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = − 𝜆2 𝑦2 𝜆2 ( 𝑥2 − 𝑥𝑦 − 𝑦2) = 𝑦2 𝑥2 − 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.

Chapter 9 Class 12 Differential Equations

Serial order wise

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