Differential Equations

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 )

The solution curve of the differential equation y dxdy=x ( e x- e y+1 ), x>0, y>0 passing through the point (e, 1) is

1. | e yx |=y2
2. | e yx |=x
3. | e xy |=y
4. 2 | e xy |=y+1

The temperature T(t) of a body at time t=0 is 160 F and it decreases continuously as per the differential equation dTdt=-K(T-80), where K is a positive constant. If T(15)=120 F, then T(45) is equal to

1. 90 F
2. 85 F
3. 80 F
4. 95 F

Let y=y(x) be the solution of the differential equation 2x dx+ (e2y 2x+ x )dy=0, 0<x< 2, y( /4)=0. If y( /6)= , then e8 is equal to .

Let y=y(x) be the solution of the differential equation x dy+\2(1-x) x+x(2-x)\dx=0 such that y(0)=2. Then y(2) is equal to:

1. 2\ (2)+1\
2. 2
3. 1
4. 2\1- (2)\

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 .

Let Y=Y(X) be a curve lying in the first quadrant such that the area enclosed by the line Y-y=Y (x)(X-x) and the co-ordinate axes, where (x, y) is any point on the curve, is always -y22 Y (x)+1, Y (x) 0. If Y(1)=1, then 12 Y(2) equals .

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