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Differential Equations

Long Answer Type

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

left parenthesis 1 plus straight x squared right parenthesis space dy over dx space plus space 2 space straight x space straight y space space equals space fraction numerator 1 over denominator 1 plus straight x squared end fraction semicolon space straight y space equals space 0 space space when space straight x space equals space 1

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292. For the given differential equation, find a particular solution satisfying the given condition:
$dy over dx minus 3 space straight y space cotx space equals space sin space 2 straight x space colon space space straight y space equals space 2 space space when space space straight x space equals space straight pi over 2$

The given differential equation is

dy over dx minus 3 straight y space cotx space equals space sin space 2 straight x

Comparing it with

dy over dx plus Py space equals straight Q comma space we space get comma space space straight P space equals space minus 3 space cotx space comma space space straight Q space equals space sin space 2 straight x

integral straight P space dx space equals space minus 3 integral cot space straight x space dx space equals space minus 3 space log space sinx

therefore space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of negative 3 space log space sinx end exponent space equals space straight e to the power of log space left parenthesis sinx right parenthesis cubed end exponent space equals space left parenthesis sin space straight x right parenthesis to the power of negative 3 end exponent space equals fraction numerator 1 over denominator sin cubed straight x end fraction

therefore

solution of given 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

straight y. space fraction numerator 1 over denominator sin space cubed straight x end fraction space equals space integral sin space 2 straight x. space 1 over sin to the power of 3 straight x end exponent dx plus space straight c

fraction numerator straight y over denominator sin cubed straight x end fraction space equals space 2 space integral fraction numerator cosx over denominator sin squared straight x end fraction dx plus straight c

fraction numerator straight y over denominator sin cubed straight x end fraction space equals space 2 space integral left parenthesis sinx right parenthesis to the power of negative 2 end exponent space cosx space dx space plus space straight c

fraction numerator straight y over denominator sin cubed straight x end fraction space equals 2 fraction numerator left parenthesis sin space straight x right parenthesis to the power of negative 1 end exponent over denominator negative 1 end fraction plus straight c

fraction numerator straight y over denominator sin cubed straight x end fraction space equals space minus fraction numerator 2 over denominator sin space straight x end fraction plus straight c

straight y space equals space minus 2 space sin squared straight x plus space straight c space sin cubed straight x

...(1)
Initially, y = 2 when

straight x equals space straight pi over 2

therefore

2 space equals space minus 2 sin squared straight pi over 2 plus straight c space sin cubed. space straight pi over 2

rightwards double arrow

2 space equals space minus 2 plus straight c

open square brackets because space space sin space straight pi over 2 space equals space 1 close square brackets

rightwards double arrow

c = 4
Putting c = 4 in (1), we get,

straight y space equals negative 2 space sin squared straight x plus 4 space sin cubed straight x comma space which space is space required space solution. space

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293. Find a particular solution of the differential equation:

dy over dx plus straight y space cotx space equals space 4 straight x

cosec space straight x space left parenthesis straight x not equal to 0 right parenthesis comma space space given space that space straight y space equals space 0 space space space when space straight x space equals space straight pi over 2.

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294. Find a particular solution of the differential equation

left parenthesis straight x plus 1 right parenthesis space dy over dx space equals space 2 straight e to the power of negative straight y end exponent minus 1 comma space space

given that y = 0, when x = 0.

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295. Solve the differential equation, given that y = 1 where x = 2

straight x dy over dx plus straight y space equals space straight x cubed.

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296. Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

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297. Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

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Short Answer Type

298.

The integrating factor of the differential equation $straight x dy over dx minus straight y space equals space 2 straight x squared$ is

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Multiple Choice Questions

299.

The integrating factor of the differential equation:
$left parenthesis 1 minus straight y squared right parenthesis dx over dy plus straight y space straight x space equals space straight a space straight y space space left parenthesis negative 1 less than straight y less than 1 right parenthesis$

$fraction numerator 1 over denominator straight y squared minus 1 end fraction$
$fraction numerator 1 over denominator square root of straight y squared minus 1 end root end fraction$
$fraction numerator 1 over denominator 1 minus straight y squared end fraction$
$fraction numerator 1 over denominator 1 minus straight y squared end fraction$

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300. The general solution of a differential equation of the type

dx over dy plus straight P subscript 1 straight x space equals space straight Q subscript 1

$straight y space straight e to the power of integral straight P subscript 1 dy end exponent space equals space integral space open parentheses straight Q subscript 1 space straight e to the power of integral straight P subscript 1 dy end exponent close parentheses space dy plus straight C$
$straight y. space straight e to the power of integral straight P subscript 1 dx end exponent space equals space integral space open parentheses straight Q subscript 1 space straight e to the power of integral straight P subscript 1 dx end exponent close parentheses dx plus straight C$
$straight x. straight e to the power of integral straight P subscript 1 dy end exponent space equals space integral open parentheses straight Q subscript 1 straight e to the power of integral straight P subscript 1 dy end exponent close parentheses dy plus straight C$
$straight x. straight e to the power of integral straight P subscript 1 dy end exponent space equals space integral open parentheses straight Q subscript 1 straight e to the power of integral straight P subscript 1 dy end exponent close parentheses dy plus straight C$

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