Check sibling questions

Ex 5.7, 14 - If y = A emx + B enx, show d2y/dx2 - (m + n)

Ex 5.7, 14 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.7, 14 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Transcript

Ex 5.7, 14 If 𝑦= 〖A𝑒〗^𝑚𝑥 + 〖B𝑒〗^𝑛𝑥, show that 𝑑2𝑦/𝑑𝑥2 − (𝑚+𝑛) 𝑑𝑦/𝑑𝑥 + 𝑚𝑛𝑦 = 0 𝑦= 〖A𝑒〗^𝑚𝑥 + 〖B𝑒〗^𝑛𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦/𝑑𝑥 = (𝑑(〖A𝑒〗^𝑚𝑥 " + " 〖B𝑒〗^𝑛𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑑(〖A𝑒〗^𝑚𝑥))/𝑑𝑥 + (𝑑(〖B𝑒〗^𝑛𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = A . 𝑒^𝑚𝑥. (𝑑(𝑚𝑥))/𝑑𝑥 + B . 𝑒^𝑛𝑥 (𝑑(𝑛𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = A . 𝑒^𝑚𝑥. 𝑚 + B . 𝑒^𝑛𝑥. 𝑛 𝑑𝑦/𝑑𝑥 = 𝐴𝑚𝑒^𝑚𝑥 + 𝐵𝑛𝑒^𝑛𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑(𝐴𝑚𝑒^𝑚𝑥 " + " 𝐵𝑛𝑒^𝑛𝑥 " " )" " /𝑑𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑(𝐴𝑚𝑒^𝑚𝑥 )/𝑑𝑥 + 𝑑(𝐵𝑛𝑒^𝑛𝑥 )" " /𝑑𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝐴𝑚 𝑑(𝑒^𝑚𝑥 )/𝑑𝑥 + 𝐵𝑛 𝑑(𝑒^𝑛𝑥 )" " /𝑑𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝐴𝑚 . 𝑒^(𝑚𝑥 ). 𝑑(𝑚𝑥 )/𝑑𝑥 + 𝐵𝑛 . 𝑒^𝑛𝑥 . 𝑑(𝑛𝑥)/𝑑𝑥 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝐴𝑚𝑒^(𝑚𝑥 ) . 𝑚+𝐵𝑛𝑒^𝑛𝑥 . 𝑛 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝐴𝑚2𝑒^(𝑚𝑥 )+𝐵𝑛2𝑒^𝑛𝑥 We need to prove (𝑑^2 𝑦)/(𝑑𝑥^2 ) − (𝑚+𝑛) 𝑑𝑦/𝑑𝑥 + 𝑚𝑛𝑦 = 0 Solving LHS (𝑑^2 𝑦)/(𝑑𝑥^2 ) − (𝑚+𝑛) 𝑑𝑦/𝑑𝑥 + 𝑚𝑛𝑦 = (𝐴𝑚2𝑒^(𝑚𝑥 )+𝐵𝑛2𝑒^𝑛𝑥) − (𝑚+𝑛) (𝐴𝑚𝑒^(𝑚𝑥 )+𝐵𝑛𝑒^𝑛𝑥) + 𝑚𝑛 (𝐴𝑒^(𝑚𝑥 )+𝐵𝑒^𝑛𝑥) = 𝐴𝑚2𝑒^(𝑚𝑥 )+𝐵𝑛2𝑒^𝑛𝑥 − 𝑚(𝐴𝑚𝑒^(𝑚𝑥 )+𝐵𝑛𝑒^𝑛𝑥) − 𝑛(𝐴𝑚𝑒^(𝑚𝑥 )+𝐵𝑛𝑒^𝑛𝑥) + 𝑚𝑛 𝐴𝑒^(𝑚𝑥 )+𝑚𝑛𝐵𝑒^𝑛𝑥 = 𝐴𝑚2𝑒^(𝑚𝑥 )+𝐵𝑛2𝑒^𝑛𝑥 −𝐴𝑚2𝑒^(𝑚𝑥 )− 𝐵𝑚𝑛𝑒^𝑛𝑥 − 𝐴𝑛𝑚𝑒^(𝑚𝑥 ) + 𝐵𝑛2𝑒^𝑛𝑥+ 𝑚𝑛 𝐴𝑒^(𝑚𝑥 )+𝑚𝑛𝐵𝑒^𝑛𝑥 = 𝐴𝑚2𝑒^(𝑚𝑥 )− 𝐴𝑚2𝑒^(𝑚𝑥 ) + 𝐵𝑛2𝑒^𝑛𝑥 −𝐵𝑛2𝑒^𝑛𝑥 − 𝐵𝑚𝑛𝑒^𝑛𝑥 + 𝐵𝑚𝑛𝑒^𝑛𝑥 − 𝐴𝑛𝑚𝑒^(𝑚𝑥 ) + 𝐴𝑛𝑚𝑒^(𝑚𝑥 ) = 0 = RHS 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.