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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Ex 5.7, 14 If ๐‘ฆ= ใ€–A๐‘’ใ€—^๐‘š๐‘ฅ + ใ€–B๐‘’ใ€—^๐‘›๐‘ฅ, show that ๐‘‘2๐‘ฆ/๐‘‘๐‘ฅ2 โˆ’ (๐‘š+๐‘›) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + ๐‘š๐‘›๐‘ฆ = 0 ๐‘ฆ= ใ€–A๐‘’ใ€—^๐‘š๐‘ฅ + ใ€–B๐‘’ใ€—^๐‘›๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘(ใ€–A๐‘’ใ€—^๐‘š๐‘ฅ " + " ใ€–B๐‘’ใ€—^๐‘›๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘(ใ€–A๐‘’ใ€—^๐‘š๐‘ฅ))/๐‘‘๐‘ฅ + (๐‘‘(ใ€–B๐‘’ใ€—^๐‘›๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = A . ๐‘’^๐‘š๐‘ฅ. (๐‘‘(๐‘š๐‘ฅ))/๐‘‘๐‘ฅ + B . ๐‘’^๐‘›๐‘ฅ (๐‘‘(๐‘›๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = A . ๐‘’^๐‘š๐‘ฅ. ๐‘š + B . ๐‘’^๐‘›๐‘ฅ. ๐‘› ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐ด๐‘š๐‘’^๐‘š๐‘ฅ + ๐ต๐‘›๐‘’^๐‘›๐‘ฅ Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = ๐‘‘(๐ด๐‘š๐‘’^๐‘š๐‘ฅ " + " ๐ต๐‘›๐‘’^๐‘›๐‘ฅ " " )" " /๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐‘‘(๐ด๐‘š๐‘’^๐‘š๐‘ฅ )/๐‘‘๐‘ฅ + ๐‘‘(๐ต๐‘›๐‘’^๐‘›๐‘ฅ )" " /๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐ด๐‘š ๐‘‘(๐‘’^๐‘š๐‘ฅ )/๐‘‘๐‘ฅ + ๐ต๐‘› ๐‘‘(๐‘’^๐‘›๐‘ฅ )" " /๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐ด๐‘š . ๐‘’^(๐‘š๐‘ฅ ). ๐‘‘(๐‘š๐‘ฅ )/๐‘‘๐‘ฅ + ๐ต๐‘› . ๐‘’^๐‘›๐‘ฅ . ๐‘‘(๐‘›๐‘ฅ)/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐ด๐‘š๐‘’^(๐‘š๐‘ฅ ) . ๐‘š+๐ต๐‘›๐‘’^๐‘›๐‘ฅ . ๐‘› (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›2๐‘’^๐‘›๐‘ฅ We need to prove (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) โˆ’ (๐‘š+๐‘›) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + ๐‘š๐‘›๐‘ฆ = 0 Solving LHS (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) โˆ’ (๐‘š+๐‘›) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ + ๐‘š๐‘›๐‘ฆ = (๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›2๐‘’^๐‘›๐‘ฅ) โˆ’ (๐‘š+๐‘›) (๐ด๐‘š๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›๐‘’^๐‘›๐‘ฅ) + ๐‘š๐‘› (๐ด๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘’^๐‘›๐‘ฅ) = ๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›2๐‘’^๐‘›๐‘ฅ โˆ’ ๐‘š(๐ด๐‘š๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›๐‘’^๐‘›๐‘ฅ) โˆ’ ๐‘›(๐ด๐‘š๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›๐‘’^๐‘›๐‘ฅ) + ๐‘š๐‘› ๐ด๐‘’^(๐‘š๐‘ฅ )+๐‘š๐‘›๐ต๐‘’^๐‘›๐‘ฅ = ๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ )+๐ต๐‘›2๐‘’^๐‘›๐‘ฅ โˆ’๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ )โˆ’ ๐ต๐‘š๐‘›๐‘’^๐‘›๐‘ฅ โˆ’ ๐ด๐‘›๐‘š๐‘’^(๐‘š๐‘ฅ ) + ๐ต๐‘›2๐‘’^๐‘›๐‘ฅ+ ๐‘š๐‘› ๐ด๐‘’^(๐‘š๐‘ฅ )+๐‘š๐‘›๐ต๐‘’^๐‘›๐‘ฅ = ๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ )โˆ’ ๐ด๐‘š2๐‘’^(๐‘š๐‘ฅ ) + ๐ต๐‘›2๐‘’^๐‘›๐‘ฅ โˆ’๐ต๐‘›2๐‘’^๐‘›๐‘ฅ โˆ’ ๐ต๐‘š๐‘›๐‘’^๐‘›๐‘ฅ + ๐ต๐‘š๐‘›๐‘’^๐‘›๐‘ฅ โˆ’ ๐ด๐‘›๐‘š๐‘’^(๐‘š๐‘ฅ ) + ๐ด๐‘›๐‘š๐‘’^(๐‘š๐‘ฅ ) = 0 = RHS Hence proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.