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

If y = ae^2x + be^βˆ’x , then show that y'' - y' - 2y = 0 - Teachoo

Question 22 - CBSE Class 12 Sample Paper for 2020 Boards - Part 2

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Question 22 If y = ae2x + beβˆ’x , then show that (𝑑^2 𝑦)/(𝑑π‘₯^2 ) βˆ’ 𝑑𝑦/𝑑π‘₯ βˆ’ 2y = 0 Given 𝑦=π‘Žπ‘’^2π‘₯+𝑏𝑒^(βˆ’π‘₯) Now, 𝑑𝑦/𝑑π‘₯=2π‘Žπ‘’^2π‘₯βˆ’π‘π‘’^(βˆ’π‘₯) And (𝑑^2 𝑦)/(𝑑π‘₯^2 )=4π‘Žπ‘’^2π‘₯+𝑏𝑒^(βˆ’π‘₯) Now, We need to show (𝑑^2 𝑦)/(𝑑π‘₯^2 ) βˆ’ 𝑑𝑦/𝑑π‘₯ βˆ’ 2y = 0 Solving LHS (𝑑^2 𝑦)/(𝑑π‘₯^2 ) βˆ’ 𝑑𝑦/𝑑π‘₯ βˆ’ 2y = (4π‘Žπ‘’^2π‘₯+𝑏𝑒^(βˆ’π‘₯)) – (2π‘Žπ‘’^2π‘₯βˆ’π‘π‘’^(βˆ’π‘₯)) – 2(π‘Žπ‘’^2π‘₯+𝑏𝑒^(βˆ’π‘₯)) = 4π‘Žπ‘’^2π‘₯+𝑏𝑒^(βˆ’π‘₯) – 2π‘Žπ‘’^2π‘₯+𝑏𝑒^(βˆ’π‘₯) – 2π‘Žπ‘’^2π‘₯βˆ’2𝑏𝑒^(βˆ’π‘₯) = (4π‘Žπ‘’^2π‘₯ "– " 2π‘Žπ‘’^2π‘₯ "– " 2π‘Žπ‘’^2π‘₯)+(𝑏𝑒^(βˆ’π‘₯) + 𝑏𝑒^(βˆ’π‘₯) βˆ’2𝑏𝑒^(βˆ’π‘₯)) = 0 + 0 = 0 Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.