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Differential Equations

Question

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Solve the differential equation:
$straight x dy over dx plus 2 straight y space equals straight x squared.$

Solution

The given differential equation is

straight x dy over dx plus 2 straight y space equals straight x squared space space or space space dy over dx plus 2 over straight x straight y space equals space straight x

Comparing it with

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

therefore space space space space integral straight P space dx space equals space integral 2 over straight x dx space equals space 2 space log space straight x space equals space log space straight x squared

therefore space 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 log space straight x squared end exponent space equals space straight x squared space

open square brackets because space space straight e to the power of log space straight x end exponent space equals space straight x close square brackets

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 space straight Q. space space straight e to the power of integral space straight P space dx end exponent dx space plus straight c

or

straight y. space straight x squared space equals space integral straight x. space straight x squared space dx plus straight c space space space space or space space space space straight x squared straight y space equals space integral straight x cubed space dx plus straight c

or

straight x squared straight y space equals space straight x to the power of 4 over 4 plus straight c space space or space space straight y space equals space 1 fourth straight x squared plus cx squared

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