Differential Equations

Solve the differential equation, given that y = 1 where x = 2 - Wired Faculty

Question

Solve the differential equation, given that y = 1 where x = 2
$straight x dy over dx plus straight y space equals space straight x cubed.$

Solution

The given differential equation is
$straight x dy over dx plus straight y space equals space straight x cubed$
or $dy over dx plus 1 over straight x straight y space equals space straight x squared$
Comparing it with $dy over dx plus Py space equals space straight Q comma space space space we space get comma space space space straight P space equals space 1 over straight x comma space space straight Q space equals space straight x squared$
$integral straight P space dx space equals space integral 1 over straight x dx space equals space log space straight x$
$therefore space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space straight x end exponent space equals space straight x$
$therefore$ solution of given differential equation is
$ye space to the power of integral straight P space dx end exponent space equals space integral space straight Q. space straight e to the power of integral straight P space dx end exponent space dx space plus straight c space space space space or space space space straight y. space straight x space equals space integral straight x squared. space space straight x space dx space plus space straight c$
or $xy equals space integral straight x cubed dx plus straight c space space or space space straight x space straight y space equals space straight x to the power of 4 over 4 plus straight c space space space or space space space straight y space equals space straight x cubed over 4 plus straight c over straight x$
Now $straight y space equals space 1 space space space space when space straight x space equals space 2$
$therefore space space space space space space space space space space space 1 space equals space 8 over 4 plus straight c over 2 space space space or space space space 1 space equals space 2 plus straight c over 2 space space space space space space rightwards double arrow space space space space straight c space equals space minus 2$
$therefore space space$ solution is $straight y space equals space straight x cubed over 4 minus straight x over 2.$

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Differential Equations

Solve the differential equation, given that y = 1 where x = 2
$straight x dy over dx plus straight y space equals space straight x cubed.$

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