Question

Solve:
$2 straight x dy over dx plus straight y space equals space 6 straight x cubed space space or space space space dy over dx plus fraction numerator straight y over denominator 2 straight x end fraction equals 3 straight x squared.$

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Solution

The given differential equation is

2 straight x dy over dx plus straight y space equals space 6 straight x cubed

dy over dx plus fraction numerator 1 over denominator 2 straight x end fraction straight y space equals space 3 straight x squared

Comparing it with

dy over dx plus straight P space straight y space space equals space straight Q comma space we space get comma

straight P space equals space fraction numerator 1 over denominator 2 straight x end fraction comma space space space straight Q space equals space 3 straight x squared

integral straight P space dx space equals space 1 half integral 1 over straight x dx space equals space 1 half log space straight x space equals space log space square root of straight x therefore space space space straight e to the power of integral space straight P space dx end exponent space equals space straight e to the power of log space square root of straight x end exponent space equals space square root of straight x therefore space space space solution space is space space space space space space space space space straight y. space straight e to the power of integral straight P space dx end exponent space equals space integral space straight Q. space straight e to the power of integral straight P space dx end exponent dx space plus straight c

straight y. space square root of straight x space equals space integral 3 space straight x squared. space square root of straight x space dx space plus space straight c

therefore space space space square root of straight x space space straight y space equals space 3 space integral straight x to the power of 5 over 2 end exponent dx plus straight c space space space space or space space space straight y square root of straight x space equals space 3 fraction numerator straight x to the power of begin display style 7 over 2 end style end exponent over denominator begin display style 7 over 2 end style end fraction plus straight c

straight y square root of straight x space equals space 6 over 7 space straight x to the power of 7 over 2 end exponent plus straight c

which is required solution.

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