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Continuity And Differentiability

Question
CBSEENMA12035390
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If y = 3e2x + 2e3x, then prove that fraction numerator straight d squared straight y over denominator dx squared end fraction minus 5 dy over dx plus 6 straight y equals 0.

Solution
space space space space space space space space space space space space space space space space straight y equals 3 straight e to the power of 2 straight x end exponent plus 2ex to the power of 3 straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space space space space space space dy over dx equals 6 straight e to the power of 2 straight x end exponent plus 6ex to the power of 3 straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis and space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 12 straight e to the power of 2 straight x end exponent plus 18ex to the power of 3 straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis space space space space space space space straight L. straight H. straight S equals fraction numerator straight d squared straight y over denominator dx squared end fraction minus 5 dy over dx plus 6 straight y equals left parenthesis 12 straight e to the power of 2 straight x end exponent plus 18ex to the power of 3 straight x end exponent right parenthesis minus 5 left parenthesis 6 straight e to the power of 2 straight x end exponent plus 6ex to the power of 3 straight x end exponent right parenthesis plus 6 left parenthesis 3 straight e to the power of 2 straight x end exponent plus 2ex to the power of 3 straight x end exponent right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis right square bracket equals 12 straight e to the power of 2 straight x end exponent plus 18ex to the power of 3 straight x end exponent minus 30 straight e to the power of 2 straight x end exponent plus 30ex to the power of 3 straight x end exponent plus 18 straight e to the power of 2 straight x end exponent plus 12ex to the power of 3 straight x end exponent equals 0 equals straight R. straight H. straight S.

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