Differential Equations

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 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 |x| = |y|exy - , , N be the solution of the differential equation x dy - y dx + xy(x dy + y dx) = 0, y(1) = 2. Then + is equal to

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 .