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Differential Equations

Question

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Solve the differential equation:
$straight x dy over dx minus straight y minus 2 straight x cubed space equals space 0.$

Solution

The given differential equation is

straight x dy over dx minus straight y minus 2 straight x cubed space equals space 0

or

straight x dy over dx minus straight y space equals space 2 straight x cubed

or

dy over dx minus 1 over straight x. space straight y space equals space 2 straight x squared

Comparing it with

dy over dx plus Py space equals straight Q comma space space we space get comma space straight P space equals space minus 1 over straight x comma space space straight Q space equals space 2 straight x squared

integral Pdx 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 straight e to the power of integral Pdx end exponent space equals space straight e to the power of log space straight x 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 plus straight c

or

straight y.1 over straight x space equals space integral 2 space straight x squared. space 1 over straight x dx space plus space straight c space space space space space space or space space space straight y over straight x space equals space 2 integral straight x space dx space plus space straight c

or

straight y over straight x space equals space straight x squared plus straight c

or

straight y space equals space straight x cubed plus cx

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