Differential Equations

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

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

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 for the solution curve y = f(x) of the differential equation dydx + ( x)y = 2 + x(1 + 2 x)2, x ( - 2, 2 ), f ( 3 ) = 310, then f ( 4 ) is equal to:

1. 5 - 32 2
2. 4 - 214
3. 9 3 + 310(4 + 3)
4. 3 + 110(4 + 3)

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