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

Transcript

Ex 9.3, 2 Form a differential equation representing the given family of curves by eliminating arbitrary constants ๐‘Ž and ๐‘. ๐‘ฆ^2=๐‘Ž(๐‘^2โˆ’๐‘ฅ^2 ) ๐‘ฆ^2=๐‘Ž(๐‘^2โˆ’๐‘ฅ^2 ) ๐‘ฆ^2=๐‘Ž๐‘^2โˆ’๐‘Ž๐‘ฅ^2 Since it has two variables, we will differentiate twice โˆด Diff. Both Sides w.r.t. ๐‘ฅ 2๐‘ฆ.๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=0โˆ’2๐‘Ž๐‘ฅ 2๐‘ฆ๐‘ฆ^โ€ฒ=โˆ’2๐‘Ž๐‘ฅ ๐‘ฆ๐‘ฆโ€ฒ=โˆ’๐‘Ž๐‘ฅ (๐‘ฆ๐‘ฆ^โ€ฒ)/(โˆ’๐‘ฅ) = ๐‘Ž ๐‘Ž = (โˆ’๐‘ฆ)/๐‘ฅ ๐‘ฆโ€ฒ Now, ๐‘ฆ๐‘ฆโ€ฒ=โˆ’๐‘Ž๐‘ฅ "Again Differentiating w.r.t. " ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ.๐‘ฆ^โ€ฒ+๐‘ฆ.(๐‘‘(๐‘ฆ^โ€ฒ))/๐‘‘๐‘ฅ=โˆ’๐‘Ž ๐‘‘๐‘ฅ/๐‘‘๐‘ฅ ๐‘ฆ^โ€ฒร—๐‘ฆ^โ€ฒ+๐‘ฆร—๐‘ฆ^โ€ฒโ€ฒ=โˆ’๐‘Ž ใ€–๐‘ฆ^โ€ฒใ€—^2+๐‘ฆ๐‘ฆ^โ€ฒโ€ฒ=โˆ’((โˆ’๐‘ฆ)/๐‘ฅ ๐‘ฆโ€ฒ) ใ€–๐‘ฆ^โ€ฒใ€—^2+๐‘ฆ๐‘ฆ^โ€ฒโ€ฒ=(๐‘ฆ๐‘ฆ^โ€ฒ)/๐‘ฅ ๐‘ฅใ€–๐‘ฆ^โ€ฒใ€—^2+๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒโ€ฒ=๐‘ฆ๐‘ฆ^โ€ฒ โ€ฆ(1) ("Using Product Rule ") (From (1) ๐‘Ž= (โˆ’๐‘ฆ)/๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ) ๐’™๐’š๐’š^โ€ฒโ€ฒ+๐’™ใ€–๐’š^โ€ฒใ€—^๐Ÿโˆ’๐’š๐’š^โ€ฒ=๐ŸŽ which is the required differential equation

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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.