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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

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Transcript

Example 6 Form the differential equation representing the family of ellipses having foci on π‘₯βˆ’π‘Žπ‘₯𝑖𝑠 is center at the origin. Ellipse whose foci is on x-axis & center at origin is π‘₯^2/π‘Ž^2 +𝑦^2/𝑏^2 =1 Differentiating both sides w.r.t. π‘₯ 𝑑/𝑑π‘₯ [π‘₯^2/π‘Ž^2 +𝑦^2/𝑏^2 ]=𝑑(1)/𝑑π‘₯ 1/π‘Ž^2 Γ—(〖𝑑(π‘₯γ€—^2))/𝑑π‘₯+1/𝑏^2 Γ—(〖𝑑(𝑦〗^2))/𝑑π‘₯=0 Since it has two variables, we will differentiate twice π‘₯^2/π‘Ž^2 +𝑦^2/𝑏^2 =1 1/π‘Ž^2 Γ—2π‘₯+1/𝑏^2 Γ—(2𝑦 . 𝑑𝑦/𝑑π‘₯)=0 2π‘₯/π‘Ž^2 +2𝑦/𝑏^2 𝑑𝑦/𝑑π‘₯=0 2𝑦/𝑏^2 𝑑𝑦/𝑑π‘₯=(βˆ’2π‘₯)/γ€– π‘Žγ€—^2 𝑦/𝑏^2 𝑑𝑦/𝑑π‘₯=(βˆ’π‘₯)/γ€– π‘Žγ€—^2 𝑦/π‘₯ 𝑑𝑦/𝑑π‘₯= (βˆ’π‘^2)/γ€– π‘Žγ€—^2 𝑦/π‘₯ 𝑦^β€²= (βˆ’π‘^2)/γ€– π‘Žγ€—^2 Again differentiating both sides 𝑑(𝑦/π‘₯)/𝑑π‘₯. 𝑦^β€²+𝑦/π‘₯ (𝑑(𝑦^β€²))/𝑑π‘₯=𝑑/𝑑π‘₯ ((βˆ’ 𝑏^2)/( π‘Ž^2 )) [𝑑𝑦/𝑑π‘₯ . π‘₯ βˆ’ 𝑦 .𝑑π‘₯/𝑑π‘₯]/π‘₯^2 𝑦^β€² +𝑦/π‘₯ ×𝑦′′=0 [𝑦^β€² π‘₯ βˆ’ 𝑦]/π‘₯^2 𝑦^β€² +𝑦/π‘₯×𝑦′′=0 Multiplying x2 both sides π‘₯^2Γ—[𝑦^β€² π‘₯ βˆ’ 𝑦]/π‘₯^2 𝑦^β€² +π‘₯^2×𝑦/π‘₯×𝑦′′=π‘₯^2Γ—0 [𝑦^β€² π‘₯βˆ’π‘¦] 𝑦^β€²+π‘₯𝑦𝑦^β€²β€²=0 γ€–γ€–π‘₯𝑦〗^β€²γ€—^2βˆ’π‘¦π‘¦^β€²+π‘₯𝑦𝑦^β€²β€²=0 π‘₯𝑦𝑦^β€²β€²+γ€–γ€–π‘₯𝑦〗^β€²γ€—^2βˆ’π‘¦π‘¦^β€²=0 π’™π’š (𝒅^𝟐 π’š)/(𝒅𝒙^𝟐 ) +𝒙(π’…π’š/𝒅𝒙)^πŸβˆ’π’š π’…π’š/𝒅𝒙=𝟎 is the required differential equation

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.