Check sibling questions

Ex 9.5, 12 - Find particular solution: x2 dy + (xy + y2) dx = 0 - Ex 9.5

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


Transcript

Ex 9.5, 12 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : 𝑥﷮2﷯𝑑𝑦+ 𝑥𝑦+ 𝑦﷮2﷯﷯ 𝑑𝑥=0;𝑦=1 When 𝑥=1 The differential equation can be written 𝑎s 𝑥﷮2﷯𝑑𝑦 = −(xy + y2) dx 𝑑𝑦﷮𝑑𝑥﷯ = − 𝑥𝑦 + 𝑦﷮2﷯﷯﷮ 𝑥﷮2﷯﷯ Let F(x, y) = 𝑑𝑦﷮𝑑𝑥﷯ = − 𝑥𝑦 + 𝑦﷮2﷯﷯﷮ 𝑥﷮2﷯﷯ Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = − 𝜆𝑥𝜆𝑦 + 𝜆﷮2﷯ 𝑦﷮2﷯﷯﷮ 𝜆﷮2﷯𝑥﷮2﷯﷯ = − 𝜆﷮2﷯ 𝑥𝑦 + 𝑦﷮2﷯﷯﷮ 𝜆﷮2﷯𝑥﷮2﷯﷯= − 𝑥𝑦 + 𝑦﷮2﷯﷯﷮ 𝑥﷮2﷯﷯ = 𝜆° F(x, y) = F(x, y) = − 𝑥𝑦 + 𝑦﷮2﷯﷯﷮ 𝑥﷮2﷯﷯ ∴ F(x, y) is a homogenous function of degree zero Putting y = vx Diff w.r.t. x 𝑑𝑦﷮𝑑𝑥﷯ = x 𝑑𝑣﷮𝑑𝑥﷯ + v Putting value of 𝑑𝑦﷮𝑑𝑥﷯ and y = vx in (1) 𝑑𝑦﷮𝑑𝑥﷯ = − 𝑥𝑦 + 𝑦﷮2﷯﷯﷮ 𝑥﷮2﷯﷯ v + 𝑑𝑣﷮𝑑𝑥﷯ = − 𝑥 𝑣𝑥﷯ + 𝑣𝑥﷯﷮2﷯﷯﷮ 𝑥﷮2﷯﷯ v + 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = −( 𝑥﷮2﷯𝑣 + 𝑥﷮2﷯ 𝑣﷮2﷯)﷮ 𝑥﷮2﷯﷯ + 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = − 𝑥﷮2﷯ (𝑣 + 𝑣﷮2﷯)﷮ 𝑥﷮2﷯﷯ v + 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = −(v2 + v) 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = − v2 − v − v 𝑥 𝑑𝑣﷮𝑑𝑥﷯ = − 𝑣﷮2﷯+2𝑣﷯ 𝑑𝑣﷮ 𝑣﷮2﷯ + 2𝑣﷯ = − 𝑑𝑥﷮𝑥﷯ Integrating both sides ﷮﷮ 𝑑𝑣﷮ 𝑣﷮2﷯ + 2𝑣﷯﷯ = − ﷮﷮ 𝑑𝑥﷮𝑥﷯﷯ ﷮﷮ 𝑑𝑣﷮ 𝑣﷮2﷯ + 2𝑣﷯﷯ = − log x + log c ﷮﷮ 𝑑𝑣﷮ (𝑣﷮2﷯ + 2𝑣 + 1) − 1﷯﷯ = − log x + log c ﷮﷮ 𝑑𝑣﷮ 𝑣 + 1﷯﷮2﷯ − 1﷮2﷯﷯﷯ = − log x + log c 1﷮2﷯ log 𝑣 + 1 − 1﷮𝑣 + 1 + 1﷯ = − log x + log c 1﷮2﷯ log 𝑣﷮𝑣 + 2﷯ = − log x + log c log ﷮𝑣﷯﷮ ﷮𝑣 + 2﷯﷯ = − log x + log C log ﷮𝑣﷯﷮ ﷮𝑣 + 2﷯﷯ + log x = log C log 𝑥 ﷮𝑣﷯﷮ ﷮𝑣 + 2﷯﷯ = log C 𝑥 ﷮𝑣﷯﷮ ﷮𝑣 + 2﷯﷯ = C Putting value of v i.e 𝑦﷮𝑥﷯ 𝑥 ﷮ 𝑦﷮𝑥﷯﷯﷮ ﷮ 𝑦﷮𝑥﷯ + 2﷯﷯ = C ﷮ 𝑥﷮2﷯ × 𝑦﷮𝑥﷯﷯﷮ ﷮ 𝑦﷮𝑥﷯ + 2﷯﷯ = C ﷮𝑥𝑦﷯﷮ ﷮ 𝑦 + 2𝑥﷮𝑥﷯﷯﷯ = C 𝑥 ﷮𝑦﷯﷮ ﷮𝑦 + 2𝑥﷯﷯ = C 𝑥 ﷮𝑦﷯ = C ﷮𝑦+2𝑥﷯ Squaring both sides x2y = c2(y + 2x) Put x = 1 & y = 1 12(1) = C2(1 + 2) 1 = 3C2 C2 = 1﷮3﷯ Putting value in (2) x2 y = 1﷮3﷯(y + 2x) 3x2y = y + 2x y + 2x = 3x2y

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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.