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

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

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

Let y = y(x) be the solution of the differential equation (1 + y2)e xdx + 2 x (1 + e2 x )dy = 0, y(0) = 1. Then y ( 4 ) is equal to

1. 1e2
2. 2e2
3. 2e
4. 1e

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 )

Suppose the solution of the differential equation dydx = (2 + )x - y + 2 x - 2 y - ( - 4 ) represents a circle passing through origin. Then the radius of this circle is :

1. 17
2. 2
3. 172
4. 12

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

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

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 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 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

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

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 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)\

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 ( 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 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 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