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)


The given differential equation is
                  dy over dx plus 3 straight y space equals space straight e to the power of mx
Comparing it with dy over dx plus Py space equals space straight Q comma space space we space get space straight P space equals space 3 comma space space straight Q space equals space straight e to the power of mx
therefore space space space space space space space space space straight e to the power of integral space straight p space dx end exponent space equals straight e to the power of 3 space integral space dx end exponent space space equals space straight e to the power of 3 straight x end exponent
therefore space space space space solution space is
                  straight y space straight e to the power of integral straight P space dx end exponent space equals integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c
or                straight y space straight e to the power of 3 straight x end exponent space equals space integral straight e to the power of mx. space straight e to the power of 3 straight x end exponent dx plus straight c
or              straight y space straight e to the power of 3 straight x end exponent space equals space integral straight e to the power of left parenthesis straight m plus 3 right parenthesis straight x end exponent dx space equals space straight c

or           straight y space straight e to the power of 3 straight x end exponent space equals space fraction numerator straight e to the power of left parenthesis straight m plus 3 right parenthesis straight x end exponent over denominator straight m plus 3 end fraction plus straight c comma space space space if space straight m space plus space 3 space not equal to space 0
If   straight m plus 3 space equals space 0 comma space space space space space straight i. straight e. comma space straight m space equals negative 3 comma space then space solution space is space   
                               space space space straight y space straight e to the power of 3 straight x end exponent space equals space integral 1 space dx plus straight c
or                             straight y space straight e to the power of 3 straight x end exponent space equals space straight x plus straight c
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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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