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CBSE Class 12-science Answered

If y = sin-1 x /(1-x2)1/2  , show that (1 – x2)d2y/dx2 – 3x dy/dx – y = 0 Explain in great detail
Asked by haroonrashidgkp | 25 Aug, 2018, 07:07: PM
answered-by-expert Expert Answer
begin mathsize 16px style straight y space equals fraction numerator space sin to the power of negative 1 end exponent straight x over denominator square root of 1 space minus space straight x squared end root end fraction dy over dx space equals space straight d over dx open parentheses fraction numerator space sin to the power of negative 1 end exponent straight x over denominator square root of 1 space minus space straight x squared end root end fraction close parentheses space space space space space space equals fraction numerator square root of 1 space minus space straight x squared end root straight d over dx sin to the power of negative 1 end exponent straight x space minus sin to the power of negative 1 end exponent straight x space straight d over dx square root of bold 1 bold space bold minus bold space bold x to the power of bold 2 end root over denominator open parentheses square root of 1 space minus space straight x squared end root close parentheses squared end fraction space space space space space space equals fraction numerator square root of 1 space minus space straight x squared end root open parentheses begin display style fraction numerator 1 over denominator square root of 1 space minus space straight x squared end root end fraction end style close parentheses space minus sin to the power of negative 1 end exponent straight x space cross times begin display style 1 half end style open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of begin display style 1 half end style minus 1 end exponent straight d over dx open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space space space space space space equals fraction numerator 1 minus space begin display style fraction numerator sin to the power of negative 1 end exponent straight x over denominator 2 square root of 1 space minus space straight x squared end root end fraction end style open parentheses 0 space minus space 2 straight x close parentheses over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space space space space space space equals fraction numerator 1 space plus space begin display style fraction numerator straight x sin to the power of negative 1 end exponent straight x over denominator square root of 1 space minus space straight x squared end root end fraction end style over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space space space space space space space equals fraction numerator 1 over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space plus space fraction numerator xsin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction...... left parenthesis 1 right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space straight d over dx open parentheses dy over dx close parentheses space space space space space space space equals space straight d over dx open parentheses fraction numerator 1 over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space plus space fraction numerator xsin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction close parentheses space space space space space space equals space space straight d over dx fraction numerator 1 over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space plus space straight d over dx fraction numerator xsin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction space space space space space space equals negative fraction numerator straight d over dx open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent straight d over dx xsin to the power of negative 1 end exponent straight x space minus xsin to the power of negative 1 end exponent straight x space straight d over dx open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent over denominator open square brackets open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent close square brackets squared end fraction space space space space space equals negative fraction numerator open parentheses 0 space minus 2 straight x close parentheses over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent open square brackets straight x straight d over dx sin to the power of negative 1 end exponent straight x space plus space sin to the power of negative 1 end exponent straight x straight d over dx straight x close square brackets space minus xsin to the power of negative 1 end exponent straight x cross times space begin display style 3 over 2 end style open parentheses open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 minus 1 end exponent close parentheses space straight d over dx open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses cubed end fraction space space space space space equals fraction numerator 2 straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent open square brackets straight x space cross times begin display style fraction numerator 1 over denominator square root of bold 1 bold space bold minus bold space bold x to the power of bold 2 end root end fraction end style space plus space sin to the power of negative 1 end exponent straight x cross times 1 space close square brackets space minus xsin to the power of negative 1 end exponent straight x cross times space begin display style 3 over 2 end style open parentheses open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 minus 1 end exponent close parentheses space straight d over dx open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses cubed end fraction space space space space equals fraction numerator 2 straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent open square brackets begin display style fraction numerator straight x over denominator square root of bold 1 bold space bold minus bold space bold x to the power of bold 2 end root end fraction end style space plus space sin to the power of negative 1 end exponent straight x space close square brackets space minus xsin to the power of negative 1 end exponent straight x cross times space begin display style 3 over 2 square root of bold 1 bold space bold minus bold space bold x to the power of bold 2 end root end style open parentheses bold 0 bold space bold minus bold 2 bold x close parentheses over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses cubed end fraction space space space space equals space fraction numerator 2 straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent open square brackets begin display style fraction numerator straight x over denominator square root of bold 1 bold space bold minus bold space bold x to the power of bold 2 end root end fraction space plus space sin to the power of negative 1 end exponent straight x space end style close square brackets space plus 3 straight x squared begin display style square root of bold 1 bold space bold minus bold space bold x to the power of bold 2 end root end style sin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses cubed end fraction space space space space equals space fraction numerator 3 straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator 3 straight x squared sin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 5 over 2 end exponent end fraction space plus fraction numerator sin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent end fraction....... left parenthesis 2 right parenthesis To space prove colon space space left parenthesis 1 space – space straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction space – space 3 straight x space open parentheses dy over dx close parentheses space – space straight y space equals space 0 space space straight L. straight H. straight S space equals space space left parenthesis 1 space – space straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction space – space 3 straight x space open parentheses dy over dx close parentheses space – space straight y space space space space space space space space space space space space equals space left parenthesis 1 space – space straight x squared right parenthesis open square brackets fraction numerator 3 straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses squared end fraction space plus space fraction numerator 3 straight x squared sin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 5 over 2 end exponent end fraction space plus fraction numerator sin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of 3 over 2 end exponent end fraction close square brackets minus space 3 straight x space open square brackets fraction numerator 1 over denominator bold 1 bold space bold minus bold space bold x to the power of bold 2 end fraction space plus space fraction numerator xsin to the power of negative 1 end exponent straight x over denominator open parentheses bold 1 bold space bold minus bold space bold x to the power of bold 2 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction close square brackets space minus space fraction numerator space sin to the power of negative 1 end exponent straight x over denominator square root of 1 space minus space straight x squared end root end fraction space space space space space space space space space space space equals space 0 space space space space space space space space space space space equals space straight R. straight H. straight S Hence space proved. end style Answered by Yasmeen Khan | 28 Aug, 2018, 12:42: PM

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