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

Transcript

Ex 9.3, 8 Form the differential equation of the family of ellipses having foci on ๐‘ฆโˆ’๐‘Ž๐‘ฅ๐‘–๐‘  and center at origin. Equation of ellipse having center at origin (0, 0) & foci on y-axis is ๐‘ฅ^2/๐‘^2 +๐‘ฆ^2/๐‘Ž^2 =1 โˆด Differentiating Both Sides w.r.t. ๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ [๐‘ฅ^2/๐‘^2 +๐‘ฆ^2/๐‘Ž^2 ] = (๐‘‘(1))/๐‘‘๐‘ฅ 1/๐‘^2 [2๐‘ฅ]+1/๐‘Ž^2 [2๐‘ฆ] ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=0 2๐‘ฅ/๐‘^2 +2๐‘ฆ/๐‘Ž^2 . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=0 Since it has two variables, we will differentiate twice 2๐‘ฆ/๐‘Ž^2 ๐‘ฆโ€ฒ=(โˆ’2๐‘ฅ)/๐‘^2 ๐‘ฆ/๐‘Ž^2 ๐‘ฆโ€ฒ=(โˆ’๐‘ฅ)/๐‘^2 (๐‘ฆ/๐‘ฅ)๐‘ฆโ€ฒ=(โˆ’๐‘Ž^2)/ใ€– ๐‘ใ€—^2 (๐‘ฆ๐‘ฆ^โ€ฒ)/๐‘ฅ = (โˆ’๐‘Ž^2)/๐‘^2 Again differentiating both sides w.r.t. x ((๐‘ฆ๐‘ฆ^โ€ฒ )^โ€ฒ ๐‘ฅ โˆ’ (๐‘‘๐‘ฅ/๐‘‘๐‘ฅ)(๐‘ฆ๐‘ฆ^โ€ฒ ))/๐‘ฅ^2 =0 (๐‘ฆ๐‘ฆ^โ€ฒ )^โ€ฒ ๐‘ฅ โˆ’ (1)(๐‘ฆ๐‘ฆ^โ€ฒ )=๐ŸŽร—๐’™^๐Ÿ (๐‘ฆ๐‘ฆ^โ€ฒ )^โ€ฒ ๐‘ฅ โˆ’๐‘ฆ๐‘ฆ^โ€ฒ=๐ŸŽ (Using Quotient rule and Diff. of constant is 0) (๐’š๐’š^โ€ฒ )^โ€ฒ ๐‘ฅ โˆ’๐‘ฆ๐‘ฆ^โ€ฒ=0 (๐’š^โ€ฒ ๐’š^โ€ฒ+๐’š๐’šโ€ฒโ€ฒ)๐‘ฅ โˆ’๐‘ฆ๐‘ฆ^โ€ฒ=0 (ใ€–๐‘ฆ^โ€ฒใ€—^2+๐‘ฆ๐‘ฆโ€ฒโ€ฒ)๐‘ฅ โˆ’๐‘ฆ๐‘ฆ^โ€ฒ=0 ๐‘ฅใ€–๐‘ฆ^โ€ฒใ€—^2+๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒโ€ฒโˆ’๐‘ฆ๐‘ฆ^โ€ฒ=0 ๐’™๐’š๐’š^โ€ฒโ€ฒ+๐’™ใ€–๐’š^โ€ฒใ€—^๐Ÿโˆ’๐’š๐’š^โ€ฒ=๐ŸŽ (Using Product rule)

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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.