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Differential Equations

Short Answer Type

271.

Solve:
$dy over dx plus 2 straight y space equals space sin space 3 straight x$

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272.

Solve:
$dy over dx plus 2 straight y space equals space sin space 5 straight x$

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273.

Solve:
$dy over dx plus 2 straight y space equals space cos space 3 straight x$

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274.

Solve:
$dy over dx plus 3 straight y space equals space sin space 2 straight x$

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275.

Solve:
$dy over dx plus 3 straight y space equals space cos space 2 straight x$

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276.
Solve:
$dy over dx plus straight y space equals space cos space 2 straight x$

The given differential equation is $dy over dx plus straight y space equals space cos space 2 straight x$
Comparing it with $dy over dx plus space straight P space straight y space equals space space straight Q comma space space we space get comma space space space straight P space equals space 1 comma space space space straight Q space equals space cos space 2 straight x$
$integral straight P space dx space equals space integral 1. space dx space equals space straight x comma space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x$
$therefore$ solution of differential equation is
   $straight y. straight e to the power of integral straight P space dx end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dx end exponent dx plus straight c$
or    $straight y. straight e to the power of straight x space space equals space integral space cos space 2 straight x. space straight e to the power of straight x space dx plus space straight c space space space or space space straight y space straight e to the power of straight x space equals space fraction numerator 1 over denominator 1 plus 4 end fraction straight e to the power of straight x left parenthesis cos space 2 straight x plus 2 space sin space 2 straight x right parenthesis space plus straight c$
or    $straight y space equals space 1 fifth left parenthesis cos space straight x plus 2 space sin space 2 straight x right parenthesis space plus space straight c space straight e to the power of straight x$