Differential Equations

Let y=y(x) be the solution of the differential equation x 1-x2 dy+ (y 1-x2-x -1x )dx=0, x (0, 1), x 1-y(x)=1. Then y ( 12 ) equals :

1. 3- 3
2. 4- 3
3. 4- 2 3
4. 3- 2 3

Let y=y(x) be the solution of the differential equation (x2-x x2-1 )dy + (y (x- x2-1 )-x )dx=0, x 1. If y(1)=1, then the greatest integer less than y ( 5 ) is .

Let f: R R be such that f(xy)=f(x)f(y), for all x, y R and f(0) 0. Let g: [1, ) R be a differentiable function such that x2g(x)= 1x (t2f(t)-tg(t) )dt. Then g(2) is equal to :

1. 138
2. 1116
3. 1532
4. 1764

Let y=y(x) be the solution of the differential equation x ( yx )dy= (y ( yx )-x )dx, y(1)= 2 and let = ( y (e12 )e12 ). Then the number of integral values of p, for which the equation x2+y2-2px+2py+ +2=0 represents a circle of radius r 6, is .

Let f(x) and g(x) be twice differentiable functions satisfying f''(x)=g''(x) for all x R, f'(1)=2g'(1)=4 and g(2)=3f(2)=9. Then f(25)-g(25) is equal to :

1. 20
2. 40
3. -20
4. -40

Let y=y(x) be the solution of the differential equation ( x)1/2 dy= ( 3x-( x)3/2y )dx, 0<x< 2, y ( 4 )= 6 25. If y ( 3 )= 45 , then 4 equals .

Let y=y(x) be the solution of the differential equation dydx= (1+x+x2 ) (1-y+y2 ), y(0)= 12. Then (2y(1)-1) is equal to :

1. 3 ( 11 36 )
2. 32 ( 11 312 )
3. 3 ( 11 312 )
4. 32 ( 11 36 )

Let y=y(x) be the solution of the differential equation : dydx+ ( 6x2+ (3x2+2x3+4 )e-2x (x3+2 ) (2+e-2x ) )y=2+e-2x, x (-1, 2), satisfying y(0)= 32. If y(1)= (2+e-2 ), then is equal to :

1. 138
2. 613
3. 1213
4. 1312

Let f be a twice differentiable function such that f(x)= 0x (t-x) dt- 0x f(t) t dt, x (- 2, 2 ). Then f'' ( 6 )+12f' (- 6 )+f ( 6 ) is equal to .

If the curve y=f(x) passes through the point (1, e) and satisfies the differential equation dy=y (2+ e x )dx, x>0, then f(e) is equal to :

1. ee
2. ee2
3. e2e
4. e2e

Let y=y(x) be the solution curve of the differential equation (1+ x) dydx+(y+1) x=0, y(0)=0. If the curve y=y(x) passes through the point ( , -12 ), then a value of is :

1. 6
2. 4
3. 3
4. 2

Let x=x(y) be the solution of the differential equation 2y2 dxdy-2xy+x2=0, y>1, x(e)=e. Then x (e2 ) is equal to:

1. 32e2
2. 23e2
3. e2
4. 2e2

Let y = y(x) be the solution of the differential equation x dydx - 2y = x3 (2 - x3 ) 2 y, x 0. If y(2) = 0, then (y(1)) is equal to

1. - 74
2. - 34
3. 34
4. 74

Let y = y(x) be the solution of the differential equation x dydx - y = x2 x, x (0, ). If y ( 2 ) = 2, then 6y ( 6 ) - 8y ( 4 ) is equal to :

1. -3
2. 3
3. -

Let a differentiable function f satisfy the equation 036 f ( tx36 )dt = 4 f(x). If y = f(x) is a standard parabola passing through the points (2, 1) and (-4, ), then is equal to .

Let y = y(x) be a differentiable function in the interval (0, ) such that y(1) = 2, and t x ( t2y(x) - x2y(t)x - t ) = 3 for each x > 0. Then 2y(2) is equal to :

1. 27
2. 18
3. 23
4. 12

Let y = y(x) be the solution of the differential equation x4 dy + (4x3y + 2 x) dx = 0, x > 0, y ( 2 ) = 0. Then 4 y ( 3 ) is equal to :

1. 92
2. 72
3. 64
4. 81

Let f be a twice differentiable non-negative function such that (f(x))2 = 25 + 0x ((f(t))2 + (f'(t))2 )dt. Then the mean of f( e(1)), f( e(2)), , f( e(625)) is equal to .

If the solution curve y = f(x) of the differential equation (x2 - 4)y' - 2xy + 2x(4 - x2)2 = 0, x > 2, passes through the point (3, 15), then the local maximum value of f is

Let the solution curve of the differential equation x dy - y dx = x2 + y2 dx, x > 0, y(1) = 0, be y = y(x). Then y(3) is equal to

1. 6
2. 4
3. 1
4. 2

Let f : [1, ) R be a differentiable function. If 6 1x f(t) dt = 3xf(x) + x3 - 4 for all x 1, then the value of f(2) - f(3) is :

1. 4
2. 3
3. -4
4. -3

If y = y(x) satisfies the differential equation 16 ( 9x + x x ) (4 + 9 + x ) y dy = (1 + 2 y) dx, x > 0 and y(256) = 2, y(49) = , then 2 is equal to :

1. 2 2 - 1
2. 2 - 1
3. 2( 2 - 1)
4. 3( 2 - 1)

Let y = y(x) be the solution curve of the differential equation (1 + x2)dy + (y - -1 x)dx = 0, y(0) = 1. Then the value of y(1) is :

1. 4e /4 - 2 - 1
2. 2e /4 + 4 - 1
3. 4e /4 + 2 - 1
4. 2e /4 - 4 - 1

Let y = y(x) be the solution of the differential equation x dydx - 2y = 2 + 3 x, x (- 2, 2 ), y(0) = - 74. Then y ( 6 ) is equal to :

1. - 52
2. -3 2 - 7
3. - 54
4. -3 3 - 7

Let f(x) = x - 1 and g(x) = ex for x R. If dydx = (e-2 x g(f(f(x))) - y x ), y(0) = 0, then y(1) is

1. 1-e3e4
2. e-1e4
3. 1-e2e4
4. 2e-1e3

Let y = y(x) be the solution curve of the differential equation x (x2 + ex )dy + (ex(x - 2)y - x3 )dx = 0, x > 0, passing through the point (1, 0). Then y(2) is equal to :

1. 22 + e2
2. 44 - e2
3. 44 + e2
4. 22 - e2

Let y = y(x) be the solution of the differential equation (x2 + 1)y' - 2xy = (x4 + 2x2 + 1) x, y(0) = 1. Then -33 y(x) dx is :

1. 36
2. 24
3. 18
4. 30

Let f : [0, ) R be a differentiable function such that f(x) = 1 - 2x + 0x ex-t f(t) dt for all x [0, ). Then the area of the region bounded by y = f(x) and the coordinate axes is

1. 5
2. 2
3. 2
4. 12

If a curve y = y(x) passes through the point (1, 2 ) and satisfies the differential equation (7x4 y - ex cosec y ) dxdy = x5, x 1, then at x = 2, the value of y is :

1. 2e2 + e64
2. 2e2 - e64
3. 2e2 - e128
4. 2e2 + e128

Let g be a differentiable function such that 0x g(t) dt = x - 0x t g(t) dt, x 0 and let y = y(x) satisfy the differential equation dydx - y x = 2(x+1) x g(x), x [0, 2 ). If y(0) = 0, then y ( 3 ) is equal to

1. 4 3
2. 2 3
3. 2 3 3
4. 4 3 3