Misc 23 - Chapter 5 Class 12 Continuity and Differentiability

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Misc 23 If 𝑦=𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥) , – 1 ≤ 𝑥 ≤ 1, show that (1−𝑥^2 ) (𝑑^2 𝑦)/〖𝑑𝑥〗^2 − 𝑥 𝑑𝑦/𝑑𝑥 −𝑎2 𝑦 0 . 𝑦=𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥) Let 〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥=𝑡 𝑦=𝑒^𝑡 Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦/𝑑𝑥 = 𝑑(𝑒^𝑡 )/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑑(𝑒^𝑡 )/𝑑𝑥 × 𝑑𝑡/𝑑𝑡 𝑑𝑦/𝑑𝑥 = 𝑑(𝑒^𝑡 )/𝑑𝑡 × 𝑑𝑡/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑒^𝑡 × 𝑑𝑡/𝑑𝑥 Putting value of 𝑡=〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥 𝑑𝑦/𝑑𝑥 = 𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥" " ) × 𝑑(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥" " ) × 𝑎 𝑑(〖𝑐𝑜𝑠〗^(−1) 𝑥)/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥" " ) × 𝑎 ((−1)/√(1 − 𝑥^2 )) 𝑑𝑦/𝑑𝑥 = (−𝑎 𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥" " ))/√(1 − 𝑥^2 ) √(1 − 𝑥^2 ) 𝑑𝑦/𝑑𝑥 = −𝑎 𝑒^(〖𝑎 𝑐𝑜𝑠〗^(−1) 𝑥" " ) √(1 − 𝑥^2 ) 𝑑𝑦/𝑑𝑥 = −𝑎 𝑦 Differentiating again w.r.t x 𝑑(√(1 − 𝑥^2 ) 𝑑𝑦/𝑑𝑥)/𝑑𝑥 = d(−𝑎𝑦)/𝑑𝑥 𝑑(√(1 − 𝑥^2 ))/𝑑𝑥 𝑑𝑦/𝑑𝑥+√(1 − 𝑥^2 ) 𝑑( 𝑑𝑦/𝑑𝑥)/𝑑𝑥 = −𝑎 𝑑𝑦/𝑑𝑥 (−1)/(2√(1 − 𝑥^2 ))× (1−𝑥^2 )^′ 𝑑𝑦/𝑑𝑥+√(1 − 𝑥^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −𝑎 𝑑𝑦/𝑑𝑥 (−1)/(2√(1 − 𝑥^2 ))× (−2𝑥) 𝑑𝑦/𝑑𝑥+√(1 − 𝑥^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −𝑎 𝑑𝑦/𝑑𝑥 𝑥/√(1 − 𝑥^2 ) 𝑑𝑦/𝑑𝑥+√(1 − 𝑥^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −𝑎 𝑑𝑦/𝑑𝑥 Multiplying √(1 − 𝑥^2 ) both sides 𝑥 𝑑𝑦/𝑑𝑥+(√(1 − 𝑥^2 ))^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −√(1 − 𝑥^2 ) 𝑎 𝑑𝑦/𝑑𝑥 𝑥 𝑑𝑦/𝑑𝑥+(1−𝑥^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −√(1 − 𝑥^2 ) 𝑎 𝑑𝑦/𝑑𝑥 𝑥 𝑑𝑦/𝑑𝑥+(1−𝑥^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = −√(1 − 𝑥^2 ) 𝑎 ×(−𝑎𝑦)/√(1 − 𝑥^2 ) 𝑥 𝑑𝑦/𝑑𝑥+(1−𝑥^2 ) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑎^2 𝑦 Hence proved From (1) √(1 − 𝑥^2 ) 𝑑𝑦/𝑑𝑥 = −𝑎 𝑦 𝑑𝑦/𝑑𝑥 = (−𝑎𝑦)/√(1 − 𝑥^2 )