Differential Equations

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 )

Let a conic C pass through the point (4, -2) and P(x, y), x 3, be any point on C. Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, -5). If the focal distance of the point (7, 1) on C is d, then 12d equals .

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 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