Misc 23 - If y = ea cos-1 x, show (1 - x2) d2y/dx2 - x dy/dx - Finding second order derivatives- Implicit form

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  1. Class 12
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Misc 23 If 𝑦=𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯) , – 1 ≀ π‘₯ ≀ 1, show that (1βˆ’π‘₯^2 ) (𝑑^2 𝑦)/〖𝑑π‘₯γ€—^2 βˆ’ π‘₯ 𝑑𝑦/𝑑π‘₯ βˆ’π‘Ž2 𝑦 0 . 𝑦=𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯) Let γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯=𝑑 𝑦=𝑒^𝑑 Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑𝑦/𝑑π‘₯ = 𝑑(𝑒^𝑑 )/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑑(𝑒^𝑑 )/𝑑π‘₯ Γ— 𝑑𝑑/𝑑𝑑 𝑑𝑦/𝑑π‘₯ = 𝑑(𝑒^𝑑 )/𝑑𝑑 Γ— 𝑑𝑑/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^𝑑 Γ— 𝑑𝑑/𝑑π‘₯ Putting value of 𝑑=γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯" " ) Γ— 𝑑(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯)/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯" " ) Γ— π‘Ž 𝑑(γ€–π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯)/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = 𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯" " ) Γ— π‘Ž ((βˆ’1)/√(1 βˆ’ π‘₯^2 )) 𝑑𝑦/𝑑π‘₯ = (βˆ’π‘Ž 𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯" " ))/√(1 βˆ’ π‘₯^2 ) Again Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑/𝑑π‘₯ (𝑑𝑦/𝑑π‘₯) = 𝑑/𝑑π‘₯ ((βˆ’π‘Ž 𝑒^(γ€–π‘Ž π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯" " ))/√(1 βˆ’ π‘₯^2 ))

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.