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Differential Equations

Question

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Solve:
$dy over dx plus straight y over straight x equals straight x squared.$

Solution

The given differential equation is

dy over dx plus straight y over straight x equals straight x squared

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma space space straight P space equals 1 over straight x. space space straight Q space equals straight x squared

integral straight P space dx space equals space integral 1 over straight x dx space equals space log space straight x

therefore 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 space equals straight x

therefore space space space solution space of space given space differential space equation space 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 space dx plus straight c space space space or space space space straight y space. straight x space equals space integral straight x squared. space straight x space dx plus straight c

or

straight x space straight y space equals space integral straight x cubed dx plus straight c space space space space or space space space straight x space straight y space equals space straight x to the power of 4 over 4 plus straight c space space space space space or space space space straight y space equals space straight x cubed over 4 plus straight c over straight x

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