Solve: from Mathematics Differential Equations

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

The given differential equation is

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 space dx over dy space equals 1 space space space space space space space space space or space space space space space dy over dx space equals 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

or

dy over dx plus fraction numerator 1 over denominator square root of straight x end fraction straight y space equals space 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

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma

straight P space equals fraction numerator 1 over denominator square root of straight x end fraction space space straight Q space equals space 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

therefore space space space space space integral straight P space dx space equals space integral fraction numerator 1 over denominator square root of straight x end fraction dx space equals space integral straight x to the power of negative 1 half end exponent dx space equals space fraction numerator straight x to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction space equals space 2 square root of straight x

therefore space space space space 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 2 square root of straight x end exponent

therefore space space space space solution space of space differential space equation space 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 space dx end exponent dx plus straight c

or

straight y space straight e to the power of 2 square root of straight x end exponent space equals integral 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 space straight e to the power of 2 square root of straight x end exponent dx plus straight c

or

straight y space straight e to the power of 2 square root of straight x end exponent space equals space integral straight x to the power of negative 1 half end exponent dx plus straight c

or

ye to the power of 2 square root of straight x end exponent space equals space fraction numerator straight x to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction plus straight c space space space space space space space or space space space space straight y space straight e to the power of 2 square root of straight x end exponent space space equals space 2 square root of straight x plus straight c

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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

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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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