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

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

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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


The given differential equation is
         left parenthesis straight x space logx right parenthesis dy over dx plus straight y space equals space log space straight x space space or space space dy over dx plus fraction numerator 1 over denominator straight x space log space straight x end fraction straight y space equals space 1 over straight x
Comparing it with dy over dx plus straight P space straight y space equals space straight Q comma space we space get comma space straight P space equals space fraction numerator 1 over denominator straight x space logx end fraction comma space space straight Q space equals space 1 over straight x
              integral straight P space dx space equals space integral fraction numerator 1 over denominator straight x space log space straight x end fraction dx space equals space integral fraction numerator begin display style 1 over straight x end style dx over denominator log space straight x end fraction space equals space log space left parenthesis log space straight x right parenthesis
therefore space space straight e to the power of integral straight P space dx end exponent space equals straight e to the power of log space left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x
therefore    solution of differential equation is
                straight y space straight e to the power of integral Pdx end exponent space equals space integral straight Q. space straight e to the power of integral straight P end exponent dx space plus space straight c apostrophe space space space or space space straight y. space log space straight x space equals space integral 1 over straight x. space log space straight x space dx plus space straight c apostrophe

or    straight y space log space straight x space equals integral left parenthesis log space straight x right parenthesis. space 1 over straight x dx plus straight c apostrophe space space or space space straight y space log space straight x space equals space fraction numerator left parenthesis log space straight x right parenthesis squared over denominator 2 end fraction plus straight c apostrophe

or         2 straight y space log space straight x space equals space left parenthesis log space straight x right parenthesis squared plus straight c
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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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