Differential Equations

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 .

The differential equation of the family of circles passing through the origin and having centre at the line y = x is :

1. (x2 - y2 + 2xy )dx = (x2 - y2 + 2xy )dy
2. (x2 + y2 - 2xy )dx = (x2 + y2 + 2xy )dy
3. (x2 + y2 + 2xy )dx = (x2 + y2 - 2xy )dy
4. (x2 - y2 + 2xy )dx = (x2 - y2 - 2xy )dy

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 .

If the solution y=y(x) of the differential equation (x4+2x3+3x2+2x+2 )dy- (2x2+2x+3 )dx=0 satisfies y(-1)=- 4, then y(0) is equal to :

1. - 12
2. 2
3. 0
4. 4

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 (x2 + 4)2dy + (2x3y + 8xy - 2)dx = 0. If y(0) = 0, then y(2) is equal to

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

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 .

Let y=y(x) be the solution of the differential equation dydx=2x(x+y)3-x(x+y)-1, y(0)=1. Then, ( 1 2+y ( 1 2 ) )2 equals :

1. 44+ e
2. 33- e
3. 21+ e
4. 12- e

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 .

Let be a non-zero real number. Suppose f: R R is a differentiable function such that f(0)=2 and x - f(x)=1. If f (x)= f(x)+3, for all x R, then f (- e 2 ) is equal to :

1. 7
2. 9
3. 3
4. 5