I had the idea to introduce a parameter into the problem to see if I could simplify it. We are looking for real pairs (p,q) that satisfy the given equation. So, letting q=kp (where k is some unknown real variable, the parameter):
2q^3 - p^3 = pq^2 + 11
2(kp)^3 - p^3 = p(kp)^2 + 11
2(k^3)(p^3) - p^3 = (k^2)(p^3) + 11
(p^3)[2k^3 - k^2 - 1] = 11
p = [11/(2k^3 - k^2 - 1)]^(1/3)
This means that all real solutions to the given equation are expressible in the form
(p,q) = ([11/(2k^3 - k^2 - 1)]^(1/3) , k*[11/(2k^3 - k^2 - 1)]^(1/3))
Here, the parameter k can range over all possible real numbers except 1 (if we let k = 1, we get division by zero in the expressions given above). [Note: Also, we have to include the point
(0, (11/2)^(1/3))
which is the limiting value of the parametric expression as we let k approach positive or negative infinity.]
I was wondering what this solution curve looks like! I was trying to work out what it would look like with pencil and paper, but I might have to resort to the graphing calculator here...
Any other ideas on this problem? It's a good one, thanks K Sengupta!
-John
ps Alas, I have been foiled by the math symbols once again! I'm editing this comment so that it will hopefully be more readable.
Edited on June 27, 2007, 8:47 pm
some thoughts
