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CBSE Class 12-science Answered

If y = xx, prove that d2y/dx2 – 1/y(dy/dx)2 – y/x = 0 explain in great detail
Asked by haroonrashidgkp | 25 Aug, 2018, 07:11: PM
answered-by-expert Expert Answer
begin mathsize 16px style straight y space equals space straight x to the power of straight x space equals space straight e to the power of logx to the power of straight x end exponent space equals straight e to the power of xlogx space dy over dx space equals space straight d over dx open parentheses straight e to the power of xlogx close parentheses space space space space space space space equals space straight e to the power of xlogx straight d over dx xlogx space space space space space space space equals straight e to the power of xlogx open parentheses straight x straight d over dx logx plus space logx straight d over dx straight x close parentheses space space space space space space equals space straight e to the power of xlogx open parentheses straight x cross times space 1 over straight x plus space logx space cross times space 1 close parentheses space space space space space space space equals straight e to the power of xlogx open parentheses 1 space plus space logx close parentheses space space space space space space equals space straight x to the power of straight x open parentheses 1 space plus space logx close parentheses space space space space space... left parenthesis 1 right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight d over dx open parentheses dy over dx close parentheses space space space space space space equals space straight d over dx straight x to the power of straight x open parentheses 1 space plus space logx close parentheses space space space space space space equals straight x to the power of straight x straight d over dx open parentheses 1 space plus space logx close parentheses plus space open parentheses 1 space plus space logx close parentheses straight d over dx straight x to the power of straight x space space space space space space equals space straight x to the power of straight x open parentheses 0 space plus space 1 over straight x close parentheses space plus space open parentheses 1 space plus space logx close parentheses straight x to the power of straight x open parentheses 1 space plus space logx close parentheses space...... from left parenthesis 1 right parenthesis space space space space space space equals straight x to the power of straight x minus 1 end exponent space plus straight x to the power of straight x space open parentheses 1 space plus space logx close parentheses squared space space space space..... left parenthesis 2 right parenthesis To space prove colon space fraction numerator straight d squared straight y over denominator dx squared end fraction space minus space 1 over straight y open parentheses dy over dx close parentheses to the power of 2 space end exponent – space straight y over straight x space equals space 0 straight L. straight H. straight S space equals fraction numerator straight d squared straight y over denominator dx squared end fraction space minus space 1 over straight y open parentheses dy over dx close parentheses to the power of 2 space end exponent – space straight y over straight x space space space space space space space space space space equals space straight x to the power of straight x minus 1 end exponent space plus straight x to the power of straight x space open parentheses 1 space plus space logx close parentheses squared minus 1 over straight x to the power of straight x open square brackets straight x to the power of straight x open parentheses 1 space plus space logx close parentheses close square brackets squared space minus straight x to the power of straight x over straight x space space space..... space from left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space space space space space space space space space space equals space 0 space space space space space space space space space space equals space straight R. straight H. straight S Hence comma space proved. end style
Answered by Yasmeen Khan | 28 Aug, 2018, 11:25: AM

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CBSE 12-science - Maths
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