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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.3, 4 Form a differential equation representing the given family of curves by eliminating arbitrary constants ๐‘Ž and ๐‘. ๐‘ฆ= ๐‘’๏ทฎ2๐‘ฅ๏ทฏ ๐‘Ž+๐‘๐‘ฅ๏ทฏ ๐‘ฆ= ๐‘’๏ทฎ2๐‘ฅ๏ทฏ ๐‘Ž+๐‘๐‘ฅ๏ทฏ โˆด Differentiating Both Sides w.r.t. ๐‘ฅ ๐‘ฆ๏ทฎโ€ฒ๏ทฏ= ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘’๏ทฎ2๐‘ฅ๏ทฏ ๐‘Ž+๐‘๐‘ฅ๏ทฏ๏ทฏ ๐‘ฆ๏ทฎโ€ฒ๏ทฏ= ๐‘‘ ๐‘’๏ทฎ2๐‘ฅ๏ทฏ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ. ๐‘Ž+๐‘๐‘ฅ๏ทฏ+ ๐‘’๏ทฎ2๐‘ฅ ๏ทฏ ๐‘‘ ๐‘Ž + ๐‘๐‘ฅ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘ฆ๏ทฎโ€ฒ๏ทฏ= 2๐‘’๏ทฎ2๐‘ฅ๏ทฏ ๐‘Ž+๐‘๐‘ฅ๏ทฏ+ ๐‘’๏ทฎ2๐‘ฅ๏ทฏ.๐‘ ๐’š๏ทฎโ€ฒ๏ทฏ= ๐’†๏ทฎ๐Ÿ๐’™๏ทฏ ๐Ÿ๐’‚+๐Ÿ๐’ƒ๐’™+๐’ƒ๏ทฏ Again differentiating w.r.t.x ๐‘ฆ^โ€ฒ=๐‘‘/๐‘‘๐‘ฅ (๐‘’^2๐‘ฅ [2๐‘Ž+2๐‘๐‘ฅ+๐‘]) yโ€ = (๐‘‘ (๐‘’^2๐‘ฅ))/๐‘‘๐‘ฅ [2๐‘Ž+2๐‘๐‘ฅ+๐‘]+๐‘’^2๐‘ฅ (๐‘‘ [2๐‘Ž+2๐‘๐‘ฅ+๐‘])/๐‘‘๐‘ฅ yโ€ = 2๐‘’^2๐‘ฅ [2๐‘Ž+2๐‘๐‘ฅ+๐‘]+๐‘’^2๐‘ฅร—2๐‘ Putting yโ€™=๐‘’^2๐‘ฅ [2๐‘Ž+2๐‘๐‘ฅ+๐‘] yโ€ = 2yโ€™ + ๐‘’^2๐‘ฅร—2๐‘ yโ€ = 2yโ€™ + 2๐‘’^2๐‘ฅ ๐‘ yโ€ โˆ’ 2yโ€™ = 2๐‘’^2๐‘ฅ ๐‘ Also, yโ€™ โˆ’ 2y = ๐‘’^2๐‘ฅ [2๐‘Ž+2๐‘๐‘ฅ +๐‘]โˆ’2๐‘’^2๐‘ฅ (๐‘Ž+๐‘๐‘ฅ) yโ€™ โˆ’ 2y = 2a๐‘’^2๐‘ฅ+2๐‘๐‘ฅ ๐‘’^2๐‘ฅ+๐‘’^2๐‘ฅ ๐‘โˆ’2๐‘Žใ€– ๐‘’ใ€—^2๐‘ฅโˆ’2๐‘๐‘ฅ ๐‘’^2๐‘ฅ yโ€™ โˆ’ 2y = (2๐‘Žใ€– ๐‘’ใ€—^2๐‘ฅโˆ’2๐‘Žใ€– ๐‘’ใ€—^2๐‘ฅ )+(2๐‘๐‘ฅ ๐‘’^2๐‘ฅโˆ’2๐‘๐‘ฅ ๐‘’^2๐‘ฅ )+๐‘’^2๐‘ฅ ๐‘ yโ€™ โˆ’ 2y = 0 + 0 + ๐‘’^2๐‘ฅ ๐‘ yโ€™ โˆ’ 2y = ๐‘’^2๐‘ฅ ๐‘ Now ((1))/((2)) , (๐‘ฆ" โˆ’ 2๐‘ฆ)/(๐‘ฆ^(โ€ฒ ) โˆ’ 2๐‘ฆ)=(2๐‘’^2๐‘ฅ ๐‘)/(๐‘’^2๐‘ฅ ๐‘) (๐‘ฆ^โ€ฒโ€ฒ โˆ’ 2๐‘ฆ^โ€ฒ)/(๐‘ฆ^โ€ฒโˆ’2๐‘ฆ)= 2 yโ€ โˆ’ 2yโ€™ = 2(yโ€™ โˆ’ 2y) yโ€ โˆ’ 2yโ€™ = 2yโ€™ โˆ’ 4y yโ€ โˆ’ 2yโ€™ โˆ’ 2yโ€™ โˆ’ 4y = 0 yโ€ โˆ’ 4yโ€™ โˆ’ 4y = 0

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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