Differential Equations

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 .

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 y = y(x) be the solution curve of the differential equation y dydx + 2x y = x3 y, y(1) = 0. Then y( 3) is equal to:

1. 6
2. 12
3. 3
4. 4

Let y = y(x) be the solution of the differential equation (2x e x) dydx + 2y = 3x e x, x > 0 and y (e-1 ) = 0. Then, y(e) is equal to

1. - 3e
2. - 32e
3. - 23e
4. - 2e

Let y = y(x) be the solution of the differential equation (1 + x2 ) dydx + y = e -1x, y(1) = 0. Then y(0) is

1. 14 (e /2 - 1 )
2. 12 (1 - e /2 )
3. 14 (1 - e /2 )
4. 12 (e /2 - 1 )

If y = y(x) is the solution of the differential equation dydx + 2y = (2x), y(0) = 34, then y ( 8 ) is equal to :

1. e- /8
2. e /4
3. e- /4
4. e /8

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 (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

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

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 dydx= ( x)+y x( x- x x), x (0, 2 ) satisfying the condition y ( 4 )=2. Then, y ( 3 ) is

1. 3 (2+ e 3 )
2. 3 (1+2 e 3 )
3. 3 (2+ e 3 )
4. 32 (2+ e 3 )

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 .

A function y=f(x) satisfies f(x) 2x+ x- (1+ 2x )f'(x)=0 with condition f(0)=0. Then, f ( 2 ) is equal to

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