Find one-parameter families of solution curves of the following

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Differential Equations

Long 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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Short 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 = e^mx(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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Long Answer Type

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260.
Find one-parameter families of solution curves of the following differential equation:
$straight y apostrophe plus straight y space cosx space equals space straight e to the power of sin space straight x end exponent space cos space straight x$

The given differential equation is

straight y apostrophe plus straight y space cosx space equals space straight e to the power of sinx space cos space straight x space space space space space space space or space space space space space dy over dx plus left parenthesis cos space straight x right parenthesis. space straight y space equals space straight e to the power of sin space straight x end exponent space space cosx space

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space we space get space straight P space equals cosx comma space straight Q space equals space straight e to the power of sinx. space cosx

therefore space space space 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 integral space cosx space dx end exponent space equals straight e to the power of sin space 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 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 sinx space equals space integral straight e to the power of sinx space cosx. space straight e to the power of sinx dx plus straight c

or

straight y space straight e to the power of sinx space equals space integral straight e to the power of 2 space sinx end exponent space cosx space dx plus straight c

...(1)
Let I =

integral straight e to the power of 2 sinx end exponent space cos space straight x space dx

Put sinx = t,

therefore space space space cos space straight x space dx space equals dt

therefore

I =

integral straight e to the power of 2 straight t end exponent dt space equals space straight e to the power of 2 straight t end exponent over 2 space equals 1 half straight e to the power of 2 sinx end exponent

therefore space space from space left parenthesis 1 right parenthesis comma space space space ye to the power of sinx space equals space 1 half straight e to the power of 2 sinx end exponent plus straight c comma space which space is space required space solution.

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