Postby boo78 » Thu Dec 12, 2013 10:56 am
Yeah, that's why I kept thinking the solution had something to do with absolute values, but it was obvious PR didn't do it right. Since posting, I've explored the slope field and overlaying solutions in more detail, and I'm now convinced that the solution does NOT involve absolute values at all. If you vary the value of c enough, you can see solutions that cover the entire slope field. Some of the things that make it tricky are that the solutions have vertical asymptotes that vary in position depending on c, which isn't obvious from either the differential equation or initial examination of the slope field, but is obvious when you look at the cubic argument of the log. Also, the solutions about the negative y-axis only appear if you graph c-values between 0 and 4/3, when the relative maximum of the cubic is positive but the relative minimum is still negative. So the answer is that y = ln ((1/3)x^3-x^2+c) is the full family of solutions, no absolute values needed. YES!