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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

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

The given differential equation is $dy over dx minus 2 straight y space equals space cos space 3 straight x$
Comparing it with $dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma space space straight P space equals space minus 2 comma space space straight Q space equals space cos space 3 straight x$
$therefore space space space space space space space integral space straight P space dx space equals space integral left parenthesis negative 2 right parenthesis space dx space equals space minus 2 space straight x comma space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of negative 2 straight x end exponent$
$therefore$ solution of differential equation is
$straight y. space straight e to the power of negative 2 straight x end exponent space equals space integral cos space 3 straight x. space straight e to the power of negative 2 straight x end exponent space dx plus straight c$ ...(1)
Let I = $integral straight e to the power of negative 2 straight x end exponent space cos space 3 straight x space dx space equals space straight e to the power of negative 2 straight x end exponent. space fraction numerator sin space 3 straight x over denominator 3 end fraction minus integral straight e to the power of negative 2 straight x end exponent left parenthesis negative 2 right parenthesis. space space fraction numerator sin space 3 straight x over denominator 3 end fraction dx$
(Integrating by parts)
$equals space 1 third straight e to the power of negative 2 straight x end exponent space sin space 3 straight x space plus space 2 over 3 integral straight e to the power of negative 2 straight x end exponent space sin space 3 straight x space dx equals space 1 third straight e to the power of negative 2 straight x end exponent space sin space 3 straight x space plus space 2 over 3 space open square brackets straight e to the power of negative 2 straight x end exponent space space open parentheses fraction numerator negative cos space 3 straight x over denominator 3 end fraction close parentheses minus space integral straight e to the power of negative 2 straight x end exponent space left parenthesis negative 2 right parenthesis. space open parentheses fraction numerator negative cos space 3 straight x over denominator 3 end fraction close parentheses close square brackets dx equals space 1 third straight e to the power of negative 2 straight x end exponent space sin space space 3 straight x space minus space 2 over 9 straight e to the power of negative 2 straight x end exponent space cos space 3 straight x space minus 4 over 9 space integral straight e to the power of negative 2 straight x end exponent space cos space 3 straight x space dx$

$therefore space space space straight I space equals space 1 over 9 straight e to the power of negative 2 straight x end exponent left parenthesis space 3 space sin space 3 straight x space space minus 2 space cos space 3 straight x right parenthesis space minus space 4 over 9 straight I space space or space space 13 over 9 straight I space equals space 1 over 9 straight e to the power of negative 2 straight x end exponent left parenthesis 3 space sin space 3 straight x space minus space 2 space cos space 3 straight x right parenthesis$
$rightwards double arrow space space space space space straight I space equals space space 1 over 13 straight e to the power of negative 2 straight x end exponent left parenthesis 3 space sin space 3 straight x minus space 2 space cos space 3 straight x right parenthesis$

$therefore space space space space from space left parenthesis 1 right parenthesis comma space solution space is$
$ye to the power of negative 2 straight x end exponent minus 1 over 13 straight e to the power of negative 2 straight x end exponent left parenthesis 3 space sin space 3 straight x minus space 2 space cos space 3 straight x right parenthesis space plus straight c$

or $straight y equals ee to the power of 2 straight x end exponent space plus space 1 over 13 left parenthesis 3 space sin space 3 straight x space minus space 2 space cos space 3 straight x right parenthesis$