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Differential Equations

Short Answer Type

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

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

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

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

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250.

Solve:
$straight x dy over dx plus 2 straight y space equals space straight x squared log space straight x$

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