Solve the following differential equation: from Mathematics Di

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Differential Equations

Long Answer Type

261.

Find one-parameter families of solution curves of the following differential equation:
$left parenthesis 1 plus straight x squared right parenthesis space dy space plus space 2 xy space dx space equals space cot space straight x space dx space space space space left parenthesis straight x not equal to 0 right parenthesis$

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262.

Find one-parameter families of solution curves of the following differential equation:
$straight x dy over dx plus straight y minus straight x plus straight x space straight y space cot space straight x space equals space 0 space space space left parenthesis straight x not equal to 0 right parenthesis$

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263.

Solve: $open parentheses fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction minus fraction numerator straight y over denominator square root of straight x end fraction close parentheses dx over dy equals 1 comma space space space straight x not equal to space 0.$

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264. Solve the following differential equation:

dy over dx space plus 2 space straight y space tan space straight x space equals space sin space straight x

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265. For the differential equation, find the general solution:

dy over dx plus 2 straight y space equals sin space straight x

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266. Solve the following differential equation:
$sin space straight x space dy over dx plus straight y space cosx space equals space cos space straight x space sin squared straight x$

The given differential equations is

sinx dy over dx plus straight y space cosx space equals space cosx space sin squared straight x

or

dy over dx plus straight y space cotx space equals space cos space straight x space sinx

Comparing it with

dy over dx plus Py space equals space straight Q comma

we get.

straight P space equals cot space straight x comma space space space straight Q space equals space space cosx space sinx

therefore space space space space integral space straight P space dx space equals space integral space cot space straight x space dx space equals space log space sinx

therefore space space space straight I. straight F. space equals 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 sinx end exponent space equals space sin space straight x

therefore space space space solution space of space differential space equation space is

straight y. space sinx space space equals space integral space cosx. space sinx. space sinx space dx space plus space straight c space space space space open square brackets because space space 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 close square brackets

or

straight y space sinx space equals space integral sin squared straight x space cosx space dx plus straight c

or

straight y space sinx space equals space fraction numerator sin cubed straight x over denominator 3 end fraction plus straight c

or

straight y equals 1 third sin squared straight x plus straight c space cosec space straight x.

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Short Answer Type

267. Find a one parameter family of solutions of each of the following differential equation:

straight y apostrophe cos squared straight x space equals space tanx minus space straight y

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268. Find a one parameter family of solutions of each of the following differential equation:

dy over dx equals negative fraction numerator straight x plus straight y space cosx over denominator 1 plus sinx end fraction

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Long Answer Type

269.

Solve:
$straight y apostrophe minus 2 straight y space equals space cos space 3 straight x$

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Short Answer Type

270.

Solve:
$straight y apostrophe plus straight y space equals space sin space straight x$

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