Differential Equations

Let f be a twice differentiable non-negative function such that (f(x))2 = 25 + 0x ((f(t))2 + (f'(t))2 )dt. Then the mean of f( e(1)), f( e(2)), , f( e(625)) is equal to .

If the solution curve y = f(x) of the differential equation (x2 - 4)y' - 2xy + 2x(4 - x2)2 = 0, x > 2, passes through the point (3, 15), then the local maximum value of f is

Let the solution curve of the differential equation x dy - y dx = x2 + y2 dx, x > 0, y(1) = 0, be y = y(x). Then y(3) is equal to

1. 6
2. 4
3. 1
4. 2

Let f : [1, ) R be a differentiable function. If 6 1x f(t) dt = 3xf(x) + x3 - 4 for all x 1, then the value of f(2) - f(3) is :

1. 4
2. 3
3. -4
4. -3

If y = y(x) satisfies the differential equation 16 ( 9x + x x ) (4 + 9 + x ) y dy = (1 + 2 y) dx, x > 0 and y(256) = 2, y(49) = , then 2 is equal to :

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

Let y = y(x) be the solution curve of the differential equation (1 + x2)dy + (y - -1 x)dx = 0, y(0) = 1. Then the value of y(1) is :

1. 4e /4 - 2 - 1
2. 2e /4 + 4 - 1
3. 4e /4 + 2 - 1
4. 2e /4 - 4 - 1

Let y = y(x) be the solution of the differential equation x dydx - 2y = 2 + 3 x, x (- 2, 2 ), y(0) = - 74. Then y ( 6 ) is equal to :

1. - 52
2. -3 2 - 7
3. - 54
4. -3 3 - 7

Let f(x) = x - 1 and g(x) = ex for x R. If dydx = (e-2 x g(f(f(x))) - y x ), y(0) = 0, then y(1) is

1. 1-e3e4
2. e-1e4
3. 1-e2e4
4. 2e-1e3

Let y = y(x) be the solution curve of the differential equation x (x2 + ex )dy + (ex(x - 2)y - x3 )dx = 0, x > 0, passing through the point (1, 0). Then y(2) is equal to :

1. 22 + e2
2. 44 - e2
3. 44 + e2
4. 22 - e2

Let y = y(x) be the solution of the differential equation (x2 + 1)y' - 2xy = (x4 + 2x2 + 1) x, y(0) = 1. Then -33 y(x) dx is :

1. 36
2. 24
3. 18
4. 30