Differential Equations

Question

Solve:
$fraction numerator dy apostrophe over denominator dx end fraction minus straight y over straight x equals space 2 straight x squared$

Solution

The given differential equation is $dy over dx minus straight y over straight x space equals space 2 straight x squared$
Comparing it with $dy over dx plus straight P space straight y space equals space straight Q comma space we space get comma space straight P space equals negative 1 over straight x comma space space straight Q space equals space 2 straight x squared.$
$integral straight P space dx space equals space minus integral 1 over straight x dx space equals space minus log space straight x space equals space log space straight x to the power of negative 1 end exponent$
$therefore space space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of logx to the power of negative 1 end exponent end exponent space equals space straight x to the power of negative 1 end exponent space equals space 1 over straight x$
$therefore$ solution of differential equation is,
$straight y space. straight e to the power of integral straight P space dx end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dx end exponent dx space plus space straight c$
or $straight y.1 over straight x space equals space integral 2 space straight x squared. space 1 over straight x dx plus straight c space space space space or space space space straight y over straight x space equals space 2 integral straight x space dx plus straight c space space or space space straight y over straight x equals straight x squared plus straight c space space or space space straight y space equals space straight x cubed plus cx$

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Differential Equations

Solve:
$fraction numerator dy apostrophe over denominator dx end fraction minus straight y over straight x equals space 2 straight x squared$

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