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Differential Equations

Short Answer Type

241. Solve the differential equation:

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

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242.

Solve:
$straight v plus 2 straight y space equals 4 straight x$

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243.

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

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244.

Solve:
$dy over dx plus 4 straight y space equals space 5 straight x$

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245.

Solve:
$dy over dx plus straight y space equals space 1$

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246.

Solve:
$fraction numerator dy apostrophe over denominator dx end fraction minus straight y over straight x equals space 2 straight x squared$

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247.

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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Long Answer Type

248.

Solve $dy over dx plus fraction numerator 1 over denominator straight x space logx end fraction straight y space equals space 1 over straight x.$

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Short Answer Type

249.
Solve:
$straight x dy over dx plus straight y space equals space straight x space log space straight x$

The given differential equation is $straight x dy over dx plus straight y space equals space straight x space log space straight x$ .
or    $dy over dx plus 1 over straight x straight y space equals space log space straight x$
Comparing it with $dy over dx plus Py space equals space straight Q comma space space we space get comma space space straight P space equals space 1 over straight x comma space space straight Q space equals space log space straight x$
$integral straight P space dx space equals space integral 1 over straight x dx space equals space log space straight x comma space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of logx space equals straight x$
$therefore space space space$ solution of differential equation is
   $straight y. space straight e to the power of integral straight P space dx end exponent space equals space integral space straight Q. space straight e to the power of integral straight P space dx end exponent space dx space plus straight c apostrophe space space or space space straight y. space straight x space equals space integral left parenthesis log space straight x right parenthesis. space straight x space dx space plus space straight c space$
or $straight y. straight x space equals left parenthesis log space straight x right parenthesis. space straight x squared over 2 minus integral 1 over straight x. straight x squared over 2 dx plus straight c apostrophe space space space or space space straight x space straight y space equals space straight x squared over 2 log space straight x space minus space 1 half integral straight x space dx plus straight c apostrophe$
or    $straight x space straight y space equals space straight x squared over 2 log space straight x space minus straight x squared over 4 plus straight c apostrophe space space or space space 4 xy space equals space 2 straight x squared logx space minus space straight x squared plus straight c$