# Example 6 - Chapter 9 Class 12 Differential Equations

Last updated at Dec. 11, 2019 by Teachoo

Last updated at Dec. 11, 2019 by Teachoo

Transcript

Example 6 Form the differential equation representing the family of ellipses having foci on ๐ฅโ๐๐ฅ๐๐ is center at the origin. Ellipse whose foci is on x-axis & center at origin is ๐ฅ^2/๐^2 +๐ฆ^2/๐^2 =1 Differentiating both sides w.r.t. ๐ฅ ๐/๐๐ฅ [๐ฅ^2/๐^2 +๐ฆ^2/๐^2 ]=๐(1)/๐๐ฅ 1/๐^2 ร(ใ๐(๐ฅใ^2))/๐๐ฅ+1/๐^2 ร(ใ๐(๐ฆใ^2))/๐๐ฅ=0 Since it has two variables, we will differentiate twice ๐ฅ^2/๐^2 +๐ฆ^2/๐^2 =1 1/๐^2 ร2๐ฅ+1/๐^2 ร(2๐ฆ . ๐๐ฆ/๐๐ฅ)=0 2๐ฅ/๐^2 +2๐ฆ/๐^2 ๐๐ฆ/๐๐ฅ=0 2๐ฆ/๐^2 ๐๐ฆ/๐๐ฅ=(โ2๐ฅ)/ใ ๐ใ^2 ๐ฆ/๐^2 ๐๐ฆ/๐๐ฅ=(โ๐ฅ)/ใ ๐ใ^2 ๐ฆ/๐ฅ ๐๐ฆ/๐๐ฅ= (โ๐^2)/ใ ๐ใ^2 ๐ฆ/๐ฅ ๐ฆ^โฒ= (โ๐^2)/ใ ๐ใ^2 Again differentiating both sides ๐(๐ฆ/๐ฅ)/๐๐ฅ. ๐ฆ^โฒ+๐ฆ/๐ฅ (๐(๐ฆ^โฒ))/๐๐ฅ=๐/๐๐ฅ ((โ ๐^2)/( ๐^2 )) [๐๐ฆ/๐๐ฅ . ๐ฅ โ ๐ฆ .๐๐ฅ/๐๐ฅ]/๐ฅ^2 ๐ฆ^โฒ +๐ฆ/๐ฅ ร๐ฆโฒโฒ=0 [๐ฆ^โฒ ๐ฅ โ ๐ฆ]/๐ฅ^2 ๐ฆ^โฒ +๐ฆ/๐ฅร๐ฆโฒโฒ=0 Multiplying x2 both sides ๐ฅ^2ร[๐ฆ^โฒ ๐ฅ โ ๐ฆ]/๐ฅ^2 ๐ฆ^โฒ +๐ฅ^2ร๐ฆ/๐ฅร๐ฆโฒโฒ=๐ฅ^2ร0 [๐ฆ^โฒ ๐ฅโ๐ฆ] ๐ฆ^โฒ+๐ฅ๐ฆ๐ฆ^โฒโฒ=0 ใใ๐ฅ๐ฆใ^โฒใ^2โ๐ฆ๐ฆ^โฒ+๐ฅ๐ฆ๐ฆ^โฒโฒ=0 ๐ฅ๐ฆ๐ฆ^โฒโฒ+ใใ๐ฅ๐ฆใ^โฒใ^2โ๐ฆ๐ฆ^โฒ=0 ๐๐ (๐ ^๐ ๐)/(๐ ๐^๐ ) +๐(๐ ๐/๐ ๐)^๐โ๐ ๐ ๐/๐ ๐=๐ is the required differential equation

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Chapter 9 Class 12 Differential Equations

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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.