If y = ae 2x + be −x , then show that (d 2 y)/(dx 2 ) − dy/dx − 2y = 0

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  1. Class 12
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Transcript

Question 22 If y = ae2x + beโˆ’x , then show that (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ’ 2y = 0 Given ๐‘ฆ=๐‘Ž๐‘’^2๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ) Now, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=2๐‘Ž๐‘’^2๐‘ฅโˆ’๐‘๐‘’^(โˆ’๐‘ฅ) And (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 )=4๐‘Ž๐‘’^2๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ) Now, We need to show (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ’ 2y = 0 Solving LHS (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ’ 2y = (4๐‘Ž๐‘’^2๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ)) โ€“ (2๐‘Ž๐‘’^2๐‘ฅโˆ’๐‘๐‘’^(โˆ’๐‘ฅ)) โ€“ 2(๐‘Ž๐‘’^2๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ)) = 4๐‘Ž๐‘’^2๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ) โ€“ 2๐‘Ž๐‘’^2๐‘ฅ+๐‘๐‘’^(โˆ’๐‘ฅ) โ€“ 2๐‘Ž๐‘’^2๐‘ฅโˆ’2๐‘๐‘’^(โˆ’๐‘ฅ) = (4๐‘Ž๐‘’^2๐‘ฅ "โ€“ " 2๐‘Ž๐‘’^2๐‘ฅ "โ€“ " 2๐‘Ž๐‘’^2๐‘ฅ)+(๐‘๐‘’^(โˆ’๐‘ฅ) + ๐‘๐‘’^(โˆ’๐‘ฅ) โˆ’2๐‘๐‘’^(โˆ’๐‘ฅ)) = 0 + 0 = 0 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.