/sci/ - Science & Math

How do I find A in terms of a and b? (I'm only interested in solving question a. first, please don't go further than this)

Thus far I don't know what to put on the other side of the equation, to isolate a and b. I know that [math]\psi[/math](x,0) must equal 1 in total, and for it to be in bounds for both nonzero conditions it must be less than 1 in either case.

Can I just use substitute a probability less than 1 and sort of hack together an answer? I can also deduce that a is nonzero and cannot equal b. a cannot be 1 because x cannot be equal to 1, it must be strictly less than 1 to save room for the second case, likewise neither can b be equal to 1, and since the two cannot be equal it must be that a<b.

However I foresee that a can equal x, which gives a possibility of using substitution to reduce the second expression to A(1), and placing in the left hand side of the first equation 1-A = A(an x less than a because otherwise absurdly 1-A = A)/a, and this yields A = 1/(x/a +1).

Plugging this new value for A into the second equation gives the left hand side (since a = x) = 1/2, but that's not what I'm looking for so I go and multiply both sides by (x/a +1) and then divide by (b-a) giving me (bx/a -x/a + b -a)/(b-x) and substituting a for x I get (2b-a-1)/(b-a) = A.

But since I didn't use any calculus I'm pretty sure this isn't the right answer. I can only do basic derivatives and I don't see the point in that if there aren't any exponents to kill.