The (small) challenge here is to determine what the LCD is. Since 11, 17 and 3 are all prime, we find the LCD by multiplying them together: 561.
Then 9/11= 459/561; 15/17 = 495/561; and 2/3 = 375/561.
(a) The differential equation is separable, so we separate the variables and integrate:



When x = 0, we have y = 2, so we solve for the constant C :

Then the particular solution to the DE is

We can go on to solve explicitly for y in terms of x :

(b) The curves y = x² and y = 2x - x² intersect for

and the bounded region is the set

The area of this region is

For the first digit there are 7 choices. For the second digit there are 6 choices (because we can't use the same one again). And so on, until there is only one choice for the last digit.
The number of possible codes is this:
7*6*5*4*3*2*1 = 7! = 5040