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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

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

The given differential equation is
$straight x dy over dx plus straight y space equals space straight x cubed$

or $dy over dx plus 1 over straight x straight y space equals space straight x squared$
Comparing it with $dy over dx plus Py space equals space straight Q comma space space space we space get comma space space space straight P space equals space 1 over straight x comma space space straight Q space equals space straight x squared$
$integral straight P space dx space equals space integral 1 over straight x dx space equals space log space straight x$
$therefore space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space straight x end exponent space equals space straight x$

$therefore$ solution of given differential equation is
$ye space to the power of integral straight P space dx end exponent space equals space integral space straight Q. space straight e to the power of integral straight P space dx end exponent space dx space plus straight c space space space space or space space space straight y. space straight x space equals space integral straight x squared. space space straight x space dx space plus space straight c$
or $xy equals space integral straight x cubed dx plus straight c space space or space space straight x space straight y space equals space straight x to the power of 4 over 4 plus straight c space space space or space space space straight y space equals space straight x cubed over 4 plus straight c over straight x$
Now $straight y space equals space 1 space space space space when space straight x space equals space 2$
$therefore space space space space space space space space space space space 1 space equals space 8 over 4 plus straight c over 2 space space space or space space space 1 space equals space 2 plus straight c over 2 space space space space space space rightwards double arrow space space space space straight c space equals space minus 2$
$therefore space space$ solution is $straight y space equals space straight x cubed over 4 minus straight x over 2.$

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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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