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

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250.
Solve:
$straight x dy over dx plus 2 straight y space equals space straight x squared log space straight x$

The given differential equation is

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

dy over dx plus 2 over straight x straight y space equals space straight x space logx

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 2 over straight x comma space space straight Q space equals space straight x space log space straight x

therefore space space space integral straight P space dx space equals space 2 space integral 1 over straight x dx space equals space 2 space log space straight x. space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of 2 space log space straight x end exponent space equals space straight e to the power of log space straight x squared end exponent space equals space straight x squared

Solution of given differential equation is

straight y space 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

straight y. straight x squared space equals space integral straight x space log space straight x. space straight x squared space dx plus straight c

or space space straight x squared. straight y space equals space integral left parenthesis log space straight x right parenthesis. space straight x cubed space dx space plus straight c or space space straight x squared straight y space equals space left parenthesis log space straight x right parenthesis. space straight x to the power of 4 over 4 minus integral 1 over straight x. straight x to the power of 4 over 4 dx plus straight c or space space space straight x squared straight y space equals space straight x to the power of 4 over 4 log space straight x space minus space 1 fourth integral straight x cubed space dx plus straight c or space space space straight x squared straight y space equals space straight x to the power of 4 over 4 log space straight x minus space 1 fourth. space straight x to the power of 4 over 4 plus straight c or space space space straight y space equals straight x squared over 4 log space straight x space minus space 1 over 16 straight x squared plus straight c space straight x to the power of negative 2 end exponent space is space the space required space solution. space

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Short Answer Type

Long Answer Type

Short Answer Type

250. Solve:

250.
Solve:
$straight x dy over dx plus 2 straight y space equals space straight x squared log space straight x$