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 .

Let f be a differentiable function such that 2(x+2)2f(x)-3(x+2)2=10 0x(t+2)f(t) dt, x 0. Then f(2) is equal to .

Let y=y(x) be the solution of the differential equation 2 x dydx= 2x-4y x, x (0, 2 ). If y ( 3 )=0, then y' ( 4 )+y ( 4 ) is equal to .

Let y=f(x) be the solution of the differential equation dydx+ xyx2-1= x6+4x 1-x2, -1<x<1 such that f(0)=0. If 6 -1/21/2f(x) dx=2 - then 2 is equal to .

For a differentiable function f : R R, suppose f'(x) = 3f(x) + , where R, f(0) = 1 and x - f(x) = 7. Then 9f(- e 3) is equal to .

Let |x| = |y|exy - , , N be the solution of the differential equation x dy - y dx + xy(x dy + y dx) = 0, y(1) = 2. Then + is equal to

Let a conic C pass through the point (4, -2) and P(x, y), x 3, be any point on C. Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, -5). If the focal distance of the point (7, 1) on C is d, then 12d equals .

If the solution y(x) of the given differential equation (ey + 1 ) x dx + ey x dy = 0 passes through the point ( 2, 0 ), then the value of ey ( 6 ) is equal to .

Let f be a differentiable function in the interval (0, ) such that f(1) = 1 and t x t2 f(x) - x2 f(t)t - x = 1 for each x > 0. Then 2f(2) + 3f(3) is equal to .

Let y = y(x) be the solution of the differential equation dydx + 2x (1 + x2 )2y = xe 1 (1 + x2 ); y(0) = 0. Then the area enclosed by the curve f(x) = y(x)e- 1 (1 + x2 ) and the line y - x = 4 is .

Let the solution y=y(x) of the differential equation dydx-y=1+4 x satisfy y( )=1. Then y ( 2 )+10 is equal to .

Let y = y(x) be the solution of the differential equation (x + y + 2)2 dx = dy, y(0) = -2. Let the maximum and minimum values of the function y = y(x) in [0, 3 ] be and , respectively. If (3 + )2 + 2 = + 3, , Z, then + equals .

If x=x(t) is the solution of the differential equation (t+1)dx= (2x+(t+1)4 )dt, x(0)=2, then, x(1) equals .

If dxdy= 1+x-y2y,x(1)=1, then 5x(2) is equal to .

Let y=y(x) be the solution of the differential equation 2x dx+ (e2y 2x+ x )dy=0, 0<x< 2, y( /4)=0. If y( /6)= , then e8 is equal to .

Let y=y(x) be the solution of the differential equation (1-x2 )dy= [xy+ (x3+2 ) 3 (1-x2 ) ]dx, -1<x<1, y(0)=0. If y ( 12 )= mn, m and n are co-prime numbers, then m+n is equal to .

Let Y=Y(X) be a curve lying in the first quadrant such that the area enclosed by the line Y-y=Y (x)(X-x) and the co-ordinate axes, where (x, y) is any point on the curve, is always -y22 Y (x)+1, Y (x) 0. If Y(1)=1, then 12 Y(2) equals .

If the solution curve y=y(x) of the differential equation (1+y2 ) (1+ ex )dx+x dy=0, x>0 passes through the point (1,1) and y(e)= - ( 32 ) + ( 32 ), then +2 is .

Let f(x)= r x \ 2r2 [(f(r))2-f(x)f(r) ]r2-x2-r3e f(r)r \ be differentiable in (- , 0) (0, ) and f(1)=1. Then the value of ea, such that f(a)=0, is equal to .

If the solution of the differential equation (2x+3y-2) dx+(4x+6y-7) dy=0, y(0)=3, is x+ y+3 e|2x+3y- |=6, then +2 +3 is equal to .