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Differential Equations

Long Answer Type

281. Find the general solution of the following differential equation:

dy over dx plus fraction numerator 4 straight x over denominator straight x squared plus 1 end fraction straight y space equals negative fraction numerator 1 over denominator left parenthesis straight x squared plus 1 right parenthesis cubed end fraction

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Short Answer Type

282.

Solve: $open parentheses straight x plus straight y close parentheses space dy over dx space equals space 1.$

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Long Answer Type

283.

Solve:
$open parentheses straight x plus 2 straight y cubed close parentheses space straight y apostrophe space equals space straight y comma space space straight y greater than 0$

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Short Answer Type

284.

Solve:
$straight y space dx space plus space left parenthesis straight x minus straight y right parenthesis squared space dy space equals space 0$

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Long Answer Type

285. Find the general solution of the differential equation y dx – (x + 2 y²) dy = 0.

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286. Solve the following differential equation:
(1 + y²)dx = (tan^{– 1} y – x) dy

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287. Solve the following differential equation:

square root of 1 minus straight y squared end root space dx equals left parenthesis sin to the power of negative 1 end exponent straight y space minus straight x right parenthesis space dy

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Short Answer Type

288. Find a one parameter family of solutions of each of the following differential equation e^–y sec² y dy = dx + x dy.

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Long Answer Type

289. Find the particular solution of the differential equation:
$dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared space cotx space space space left parenthesis straight x not equal to 0 right parenthesis$

The given differential equation is

dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared cotx thin space left parenthesis straight x not equal to 0 right parenthesis

given that y = 0 when x =

straight pi over 2.

dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared space cot space straight x

Comparing it with

dy over dx plus Py space equals space straight Q comma

we get,

straight P space equals space cotx comma space space straight Q space equals space 2 straight x plus straight x squared space cotx space space space integral straight P space dx space equals space integral cot space straight x space dx space equals space log space sinx. 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 sinx

Solution of differential equation is

straight y space straight e to the power of integral straight P space dx end exponent space equals space integral space straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c

or space space space space space space straight y space sinx space equals space integral left parenthesis 2 straight x plus straight x squared space cot space straight x right parenthesis space sinx space dx plus straight c or space space space space space space straight y space sinx space equals space integral 2 straight x space sinx space dx space space plus space integral straight x squared space cosx space dx or space space space space space straight y space sinx space equals space left parenthesis sinx right parenthesis. space open parentheses fraction numerator 2 straight x squared over denominator 2 end fraction close parentheses space minus space integral space cos space straight x space open parentheses fraction numerator 2 straight x squared over denominator 2 end fraction close parentheses dx plus space integral space straight x squared space cosx space dx space plus space straight c or space space space space straight y space sinx space equals space straight x squared space sinx space minus space integral straight x squared space cosx space dx space plus space integral straight x squared space cos space straight x space dx space plus space straight c or space space space space straight y space sinx space equals space straight x squared space sin space straight x space plus space straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

Now y = 0 when

straight x space equals space straight pi over 2

therefore space space space space space 0 space equals space straight pi squared over 4 space sin space straight pi over 2 plus straight c space space space space space space rightwards double arrow space space space space 0 space equals space straight pi squared over 4 cross times 1 plus space straight c space space space rightwards double arrow space space space straight c space space equals negative straight pi squared over 4

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290. For the given differential equation, find a particular solution satisfying the given condition:

dy over dx plus 2 straight y space tan space straight x space equals space sin space straight x space colon space straight y space space equals space 0 space space when space straight x space equals space straight pi over 3

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