Differential Equations · Question 1

Let y=f(x) be the real valued function defined on the interval (0, ), satisfying y(1)=0 and the differential equation x dydx=y-x3. Then which of the following statements is (are) TRUE ?

1. The function f has a local minimum at x= 1 3
2. The function f has a local maximum at x= 1 3
3. The function f is increasing in the interval (1, 2)
4. If g(x)=4x3-5x2+ 32x for x>0, then the number of elements in the set \x (0, ): f(x)=g(x)\ is 2

For all x > 0, let y1(x), y2(x), and y3(x) be the functions satisfying dy1dx - ( x)2y1 = 0, y1(1) = 5, dy2dx - ( x)2y2 = 0, y2(1) = 13, dy3dx - ( 2 - x3x3 )y3 = 0, y3(1) = 35e, respectively. Then x 0+ y1(x)y2(x)y3(x) + 2xe3x x is equal to .

Let y(x) be the solution of the differential equation x2 dydx+xy=x2+y2, x> 1e, satisfying y(1)=0. Then the value of 2 (y(e))2y(e2) is .

Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t10f(x)-x10f(t)t9-x9=1 for each x>0. Then, for all x>0, f(x) is equal to

1. 3111x- 911x10
2. 911x+ 1311x10
3. -911x+ 3111x10
4. 1311x+ 911x10