Differential Equations

If the solution y(x) of the given differential equation (ey + 1 ) x dx + ey x dy = 0 passes through the point ( 2, 0 ), then the value of ey ( 6 ) is equal to .

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 .

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

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

If the solution curve y=y(x) of the differential equation (1+y2 ) (1+ ex )dx+x dy=0, x>0 passes through the point (1,1) and y(e)= - ( 32 ) + ( 32 ), then +2 is .

If ( yx )= e|x|+ 2 is the solution of the differential equation x ( yx ) dydx=y ( yx )+x and y(1)= 3, then 2 is equal to

1. 12
2. 9
3. 4
4. 3

Let f(x)= r x \ 2r2 [(f(r))2-f(x)f(r) ]r2-x2-r3e f(r)r \ be differentiable in (- , 0) (0, ) and f(1)=1. Then the value of ea, such that f(a)=0, is equal to .

Let x=x(t) and y=y(t) be solutions of the differential equations dxdt+ax=0 and dydt+by=0 respectively, a, b . Given that x(0)=2; y(0)=1 and 3y(1)=2x(1), the value of t, for which x(t)=y(t), is :

1. 232
2. 432
3. 4 3
4. 3 4

If the solution of the differential equation (2x+3y-2) dx+(4x+6y-7) dy=0, y(0)=3, is x+ y+3 e|2x+3y- |=6, then +2 +3 is equal to .

If y=y(x) is the solution curve of the differential equation (x2-4 )dy- (y2-3y )dx=0, x>2, y(4)= 32 and the slope of the curve is never zero, then the value of y(10) equals :

1. 31+(8)1/4
2. 31-(8)1/4
3. 31-2 2
4. 31+2 2

If the solution curve, of the differential equation dydx= x+y-2x-y passing through the point (2,1) is -1 ( y-1x-1 )- 1 e ( + ( y-1x-1 )2 )= e|x-1|, then 5 + is equal to .