Differential Equations

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 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 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 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 .

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 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 f : R R be a thrice differentiable odd function satisfying f'(x) 0, f''(x) = f(x), f(0) = 0, f'(0) = 3. Then 9f( e 3) is equal to .

Let y = y(x) be the solution of the differential equation dydx + 2y 2 x = 2 2 x + 3 x 2 x such that y(0) = 54. Then 12 (y ( 4 ) - e-2 ) is equal to .

Let f : (0, ) R be a twice differentiable function. If for some a 0, 01 f( x) d = af(x), f(1) = 1 and f(16) = 18, then 16 - f' ( 116 ) is equal to .

If y = y(x) is the solution of the differential equation, 4 - x2 dydx = ( ( -1 ( x2 ) )2 - y ) -1 ( x2 ), -2 x 2, y(2) = 2 - 84, then y2(0) is equal to .