Differential Equations

Let for some function y = f(x), 0x t f(t) dt = x2 f(x), x > 0 and f(2) = 3. Then f(6) is equal to

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

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 y=y(x) be the solution of the differential equation (xy-5x2 1+x2 )dx+ (1+x2 )dy=0, y(0)=0. Then y( 3) is equal to

1. 5 32
2. 152
3. 143
4. 2 2

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 f:(0, ) be a function which is differentiable at all points of its domain and satisfies the condition x2f'(x)=2xf(x)+3, with f(1)=4. Then 2f(2) is equal to :

1. 19
2. 23
3. 29
4. 39

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 a curve y=f(x) pass through the points (0,5) and ( e 2,k). If the curve satisfies the differential equation 2(3+y)e2xdx-(7+e2x)dy=0, then k is equal to

1. 32
2. 8
3. 4
4. 16

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

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

Let x=x(y) be the solution of the differential equation y2 dx+ (x- 1y )dy=0. If x(1)=1, then x ( 12 ) is :

1. 32+e
2. 12+e
3. 3+e
4. 3-e

If x=f(y) is the solution of the differential equation (1+y2 )+ (x-2e -1y ) dydx=0, y (- 2, 2 ) with f(0)=1, then f ( 1 3 ) is equal to :

1. e /4
2. e /12
3. e /6
4. e /3