Question

Find the general solution of the following differential equation:
$straight x dy over dx minus straight a space straight y space equals space straight x plus 1 comma space space space straight x greater than 0$

Solution

The given differential equation is $straight x dy over dx minus ay space equals space straight x plus 1$
Dividing throughout by x, we get,
   $dy over dx minus straight a over straight x straight y space equals space fraction numerator straight x plus 1 over denominator straight x end fraction$ ...(1)
Comparing (1) with $dy over dx plus straight P space straight y space equals space straight Q comma$ we have, $straight P space equals negative straight a over straight x comma space space straight Q space equals fraction numerator straight x plus 1 over denominator straight x end fraction$
$therefore space space space space space integral straight P space dx equals space minus straight a integral 1 over straight x dx space equals negative straight a space logx space equals space log space straight x to the power of negative straight a end exponent therefore space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space straight x to the power of negative straight a end exponent end exponent space equals space straight x to the power of negative straight a end exponent space equals space 1 over straight x to the power of straight a$
$therefore$ solution of given equation is, $straight y. space 1 over straight x to the power of straight a space equals space integral fraction numerator straight x plus 1 over denominator straight x end fraction 1 over straight x to the power of straight a dx plus straight c$
or    $straight y over straight x to the power of straight a space equals integral open parentheses 1 over straight x to the power of straight a plus 1 over straight x to the power of straight a plus 1 end exponent close parentheses dx plus straight c$
or    $straight y over straight x to the power of straight a space equals fraction numerator straight x to the power of 1 minus straight a end exponent over denominator 1 minus straight a end fraction plus fraction numerator straight x to the power of negative straight a end exponent over denominator negative straight a end fraction plus straight c space space space space or space space space straight y space equals space fraction numerator straight x over denominator 1 minus straight a end fraction minus 1 over straight a plus cx to the power of straight a$

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Differential Equations

Find the general solution of the following differential equation:
$straight x dy over dx minus straight a space straight y space equals space straight x plus 1 comma space space space straight x greater than 0$

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