kingkongsr. said:
I just looked at the equation again, it is suppose to be: dy/dx = m*x + b - k*y^2 where m is multiplied by x. However, can both sides be multiplied by dx such that dy = (m*x + b - k*y^2) dx where the right hand side can be integrated with respect to x. Then both sides are divided by the integral of the right side such that dy/[IntegralWithRespectx(m*x + b - k*y^2)] = 1. Then the left hand side can be integrated with respect to y such that, IntegralWithRespecty[1/[IntegralWithRespectx(m*x + b - k*y^2)]] =1. I have never heard of that, but can that be done?
No, what you're trying to do there is a separation of variables, but you can't separate the y^2 term on the right hand side from the x terms, so you still have to integrate over y^2
with respect to x - effectively what you've done is converted a differential equation into an "integral equation".
It's weird that taking another derivative, as suggested above, doesn't seem to be be useful. Taking the derivative, you get:
\frac{d^2y}{dx^2} + 2ky = m
This is a linear second order homogeneous equation which can be solved in closed form, but the first derivative of the solution that equation can't be matched to mx + b. Huh... I haven't seen that before...
If you do want to try and solve it via the suggested "Riccati Equation", of which this is apparently a form of, with q_1(x) = 0, try looking at the wikipedia article:
http://en.wikipedia.org/wiki/Riccati_equation. Of course, if computers are giving Airy functions as solutions, then you're probably going to have to resort to a power series solution once you boil that down to the linear 2nd order equation as described in the article (or perhaps you can identify the resulting second order ODE with Airy's equation: u'' + xu = 0 - but probably not. I don't know; that computer solution is ugly.)