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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise

Transcript

Ex 9.3, 12 Which of the following differential equations has ๐‘ฆ=๐‘ฅ as one of its particular solution ? (A) (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )โˆ’๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=๐‘ฅ (B) (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )+๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=๐‘ฅ (C) ) (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )โˆ’๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=0 (D) (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )+๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=0 ๐‘ฆ=๐‘ฅ Differentiating both sides w.r.t. ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=1 Again differentiating both sides w.r.t. ๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )=0 Let us check each Options Option A (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) โˆ’๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ +๐‘ฅ๐‘ฆ=๐‘ฅ Putting (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = 1, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0, y = x 0โˆ’๐‘ฅ^2 (1)+๐‘ฅ(๐‘ฅ)=๐‘ฅ โˆ’๐‘ฅ^2+๐‘ฅ^2=๐‘ฅ 0=๐‘ฅ Since this is not true โˆด Option (A) is not possible Option B (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) +๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=๐‘ฅ Putting (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = 1, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0, y = x 0+๐‘ฅ(1)+๐‘ฅ(๐‘ฅ)=๐‘ฅ ๐‘ฅ+๐‘ฅ^2=๐‘ฅ ๐‘ฅ^2=0 Since this is not true โˆด Option (B) is not possible Option C (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )โˆ’๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=0 Putting (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = 1, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0, y = x 0โˆ’๐‘ฅ^2 (1)+๐‘ฅ(๐‘ฅ)=0 โˆ’๐‘ฅ^2+๐‘ฅ^2=0 0=0 Since this is true โˆด Option (C) is possible Option D (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) +๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ+๐‘ฅ๐‘ฆ=0 Putting (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = 1, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0, y = x 0+๐‘ฅ(1)+๐‘ฅ(๐‘ฅ)=0 ๐‘ฅ+๐‘ฅ^2=0 ๐‘ฅ=โˆ’๐‘ฅ^2 Since this is not true โˆด Option (D) is not possible Thus, Option C is correct

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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.