Differential Equations

Question

$If space straight y space equals space Pe to the power of ax plus Qe to the power of bx$ show that
$straight d to the power of 2 straight y end exponent over dx squared minus left parenthesis straight a plus straight b right parenthesis dy over dx plus aby space equals 0$

Solution

Given that
$straight y equals Pe to the power of ax plus Qe to the power of bx Differentiating space the space above space function space with space respect space to space straight x comma dy over dx equals Pae to the power of ax plus Qbe to the power of bx Differentiating space once space again comma space we space have comma fraction numerator straight d squared straight y over denominator dx squared end fraction equals space Pa squared straight e to the power of ax plus Qb squared straight e to the power of bx$
Let us now find $left parenthesis straight a plus straight b right parenthesis dy over dx colon$
$left parenthesis straight a plus straight b right parenthesis dy over dx equals left parenthesis straight a plus straight b right parenthesis open parentheses Pae to the power of ax plus Qbe to the power of bx close parentheses space space rightwards double arrow space left parenthesis straight a plus straight b right parenthesis dy over dx equals Pa squared straight e to the power of ax plus Qabe to the power of bx plus Pabe to the power of bx plus Qb squared straight e to the power of bx rightwards double arrow space left parenthesis straight a plus straight b right parenthesis dy over dx equals Pa squared straight e to the power of ax plus left parenthesis straight P plus straight Q right parenthesis abe to the power of bx plus Qb squared straight e to the power of bx space space space$
Also we have,
$aby space equals ab open parentheses Pe to the power of ax plus Qe to the power of bx close parentheses space equals space ab Thus comma space fraction numerator straight d squared straight y over denominator dx squared end fraction minus left parenthesis straight a plus straight b right parenthesis dy over dx plus aby equals Pa squared straight e to the power of ax plus Qb squared straight e to the power of bx minus Pa squared straight e to the power of ax minus left parenthesis straight P plus straight Q right parenthesis abe to the power of bx minus Qb squared straight e to the power of bx plus abPe to the power of ax plus abQe to the power of bx equals 0$

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Differential Equations

$If space straight y space equals space Pe to the power of ax plus Qe to the power of bx$ show that
$straight d to the power of 2 straight y end exponent over dx squared minus left parenthesis straight a plus straight b right parenthesis dy over dx plus aby space equals 0$

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