Example 6 - Family of ellipses having foci on x-axis, center


Example 6 - Chapter 9 Class 12 Differential Equations - Part 2


Example 6 - Chapter 9 Class 12 Differential Equations - Part 3

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


Example 6 Form the differential equation representing the family of ellipses having foci on ๐‘ฅโˆ’๐‘Ž๐‘ฅ๐‘–๐‘  is center at the origin. Ellipse whose foci is on x-axis & center at origin is ๐‘ฅ^2/๐‘Ž^2 +๐‘ฆ^2/๐‘^2 =1 Differentiating both sides w.r.t. ๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ [๐‘ฅ^2/๐‘Ž^2 +๐‘ฆ^2/๐‘^2 ]=๐‘‘(1)/๐‘‘๐‘ฅ 1/๐‘Ž^2 ร—(ใ€–๐‘‘(๐‘ฅใ€—^2))/๐‘‘๐‘ฅ+1/๐‘^2 ร—(ใ€–๐‘‘(๐‘ฆใ€—^2))/๐‘‘๐‘ฅ=0 Since it has two variables, we will differentiate twice ๐‘ฅ^2/๐‘Ž^2 +๐‘ฆ^2/๐‘^2 =1 1/๐‘Ž^2 ร—2๐‘ฅ+1/๐‘^2 ร—(2๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ)=0 2๐‘ฅ/๐‘Ž^2 +2๐‘ฆ/๐‘^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=0 2๐‘ฆ/๐‘^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=(โˆ’2๐‘ฅ)/ใ€– ๐‘Žใ€—^2 ๐‘ฆ/๐‘^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=(โˆ’๐‘ฅ)/ใ€– ๐‘Žใ€—^2 ๐‘ฆ/๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ= (โˆ’๐‘^2)/ใ€– ๐‘Žใ€—^2 ๐‘ฆ/๐‘ฅ ๐‘ฆ^โ€ฒ= (โˆ’๐‘^2)/ใ€– ๐‘Žใ€—^2 Again differentiating both sides ๐‘‘(๐‘ฆ/๐‘ฅ)/๐‘‘๐‘ฅ. ๐‘ฆ^โ€ฒ+๐‘ฆ/๐‘ฅ (๐‘‘(๐‘ฆ^โ€ฒ))/๐‘‘๐‘ฅ=๐‘‘/๐‘‘๐‘ฅ ((โˆ’ ๐‘^2)/( ๐‘Ž^2 )) [๐‘‘๐‘ฆ/๐‘‘๐‘ฅ . ๐‘ฅ โˆ’ ๐‘ฆ .๐‘‘๐‘ฅ/๐‘‘๐‘ฅ]/๐‘ฅ^2 ๐‘ฆ^โ€ฒ +๐‘ฆ/๐‘ฅ ร—๐‘ฆโ€ฒโ€ฒ=0 [๐‘ฆ^โ€ฒ ๐‘ฅ โˆ’ ๐‘ฆ]/๐‘ฅ^2 ๐‘ฆ^โ€ฒ +๐‘ฆ/๐‘ฅร—๐‘ฆโ€ฒโ€ฒ=0 Multiplying x2 both sides ๐‘ฅ^2ร—[๐‘ฆ^โ€ฒ ๐‘ฅ โˆ’ ๐‘ฆ]/๐‘ฅ^2 ๐‘ฆ^โ€ฒ +๐‘ฅ^2ร—๐‘ฆ/๐‘ฅร—๐‘ฆโ€ฒโ€ฒ=๐‘ฅ^2ร—0 [๐‘ฆ^โ€ฒ ๐‘ฅโˆ’๐‘ฆ] ๐‘ฆ^โ€ฒ+๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒโ€ฒ=0 ใ€–ใ€–๐‘ฅ๐‘ฆใ€—^โ€ฒใ€—^2โˆ’๐‘ฆ๐‘ฆ^โ€ฒ+๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒโ€ฒ=0 ๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒโ€ฒ+ใ€–ใ€–๐‘ฅ๐‘ฆใ€—^โ€ฒใ€—^2โˆ’๐‘ฆ๐‘ฆ^โ€ฒ=0 ๐’™๐’š (๐’…^๐Ÿ ๐’š)/(๐’…๐’™^๐Ÿ ) +๐’™(๐’…๐’š/๐’…๐’™)^๐Ÿโˆ’๐’š ๐’…๐’š/๐’…๐’™=๐ŸŽ 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 10 years. He provides courses for Maths and Science at Teachoo.