Solve: from Mathematics Differential Equations

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 Multiple Choice QuestionsLong Answer Type

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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


The given differential equation is
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 space space space or space space space dy over dx plus space fraction numerator 1 over denominator straight x space log space straight x end fraction straight y space equals space 2 over straight x squared
Comparing it with dy over dx plus Py space equals straight Q comma space we space get comma space space straight P space equals fraction numerator 1 over denominator straight x space log space straight x end fraction comma space straight Q space equals 2 over straight x squared
therefore space space space space space space 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 integral fraction numerator begin display style 1 over straight x end style over denominator log space straight x end fraction dx space equals space log space left parenthesis log space straight x right parenthesis
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 left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x.
therefore   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 space plus straight c
or             straight y space log space straight x space equals space integral 2 over straight x squared log space straight x space dx space plus space straight c

or           straight y space log space straight x space equals space 2 space integral left parenthesis log space straight x right parenthesis. space straight x squared space dx space plus straight c
or space space space straight y space log space straight x space equals space 2 open square brackets left parenthesis log space straight x right parenthesis. space fraction numerator straight x to the power of negative 1 end exponent over denominator negative 1 end fraction minus integral 1 over straight x. fraction numerator straight x to the power of negative 1 end exponent over denominator negative 1 end fraction dx close square brackets plus straight c
or space space straight y space log space straight x space equals space minus fraction numerator 2 space log space straight x over denominator straight x end fraction plus 2 space integral straight x to the power of negative 2 end exponent space dx plus straight c
or space space straight y space logx space space equals space minus fraction numerator 2 space log space straight x over denominator straight x end fraction plus 2 fraction numerator straight x to the power of negative 1 end exponent over denominator negative 1 end fraction plus straight c
or space space space straight y space logx space equals negative 2 over straight x left parenthesis 1 plus log space straight x right parenthesis space plus space straight c comma space which space is space required space solution.
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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

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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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