Differential Equations

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

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 .

The solution of the differential equation (x2 + y2 )dx - 5xy dy = 0, y(1) = 0, is :

1. |x2 - 4y2 |5 = x2
2. |x2 - 2y2 |6 = x
3. |x2 - 2y2 |5 = x2
4. |x2 - 4y2 |6 = x

The solution curve, of the differential equation 2y dydx + 3 = 5 dydx, passing through the point (0, 1) is a conic, whose vertex lies on the line :

1. 2x + 3y = -9
2. 2x + 3y = -6
3. 2x + 3y = 9
4. 2x + 3y = 6

Let 0x 1 - (y'(t))2 dt = 0x y(t) dt, 0 x 3, y 0, y(0) = 0. Then at x = 2, y'' + y + 1 is equal to

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

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 f(x) be a positive function such that the area bounded by y = f(x), y = 0 from x = 0 to x = a > 0 is e-a + 4a2 + a - 1. Then the differential equation, whose general solution is y = c1 f(x) + c2, where c1 and c2 are arbitrary constants, is

1. (8ex + 1 ) d2ydx2 - dydx = 0
2. (8ex + 1 ) d2ydx2 + dydx = 0
3. (8ex - 1 ) d2ydx2 - dydx = 0
4. (8ex - 1 ) d2ydx2 + dydx = 0