Find one-parameter families of solution curves of the following

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 Multiple Choice QuestionsLong Answer Type

251.

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

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 Multiple Choice QuestionsShort Answer Type

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252.

Find one-parameter families of solution curves of the following differential equations:
straight y apostrophe plus 3 straight y space equals straight e to the power of negative 2 straight x end exponent


The given differential equation is
                      straight y apostrophe plus 3 straight y space equals space straight e to the power of negative 2 straight x end exponent
or     dy over dx plus 3 straight y space equals space straight e to the power of negative 2 straight x end exponent
Comparing it with dy over dx plus Py space equals straight Q comma space we space get comma space straight P space equals space 3 comma space straight Q space equals space straight e to the power of negative 2 straight x end exponent
therefore space space space space space integral straight P space dx space equals space 3 space integral space 1 space dx space equals space 3 space straight x. space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of 3 straight x end exponent
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 space straight Q space straight e to the power of integral straight P space dx end exponent dx plus space straight c
or       straight y space straight e to the power of 3 straight x end exponent space equals space integral space straight e to the power of negative 2 straight x end exponent. space straight e to the power of 3 straight x end exponent space dx plus straight c
or space space space straight y space straight e to the power of 3 straight x end exponent space equals space integral straight e to the power of straight x space dx plus straight c
or space space space space straight y space straight e to the power of 3 straight x end exponent space equals space straight e to the power of straight x plus straight c
or space space space space space space space straight y space space equals space straight e to the power of negative 2 straight x end exponent plus straight c space straight e to the power of negative 3 straight x end exponent space is space required space solution.
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253.

Find one-parameter families of solution curves of the following differential equation:
 straight y apostrophe space minus space straight y space equals space cos space straight x

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254.

Find one-parameter families of solution curves of the following differential equation:
 y'+3 y = emx (m: a given real number)

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255.

Find one-parameter families of solution curves of the following differential equation:
 y' - y = cos 2x

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256.

Find one-parameter families of solution curves of the following differential equation:
x y' - y = (x+1) e-x

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257.

Find one-parameter families of solution curves of the following differential equation:
x y' + y = x4

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258.

Find one-parameter families of solution curves of the following differential equation:
x y' log x + y = log x

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259.

Find one-parameter families of solution curves of the following differential equation:
dy over dx minus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction comma space space straight y space equals space straight x squared plus 2

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 Multiple Choice QuestionsLong Answer Type

260.

Find one-parameter families of solution curves of the following differential equation:
  straight y apostrophe plus straight y space cosx space equals space straight e to the power of sin space straight x end exponent space cos space straight x

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