Really need someone's help with calculus

Question

The slope field for a differential equation is shown in the figure. Determine the general solution of this equation.

Options:

1)	y=Cx2
2)	x=Cy2
3)	y2 - x2 = C2
4)	x2 + y2 = C2

Answer 1

Hi Cheyanna B.,

I'm a little behind the times on terminology of "slope field", but the intent of the question is clear: for the types of functions (or not!) listed, what are their slopes in the various quadrants. You can eliminate (1) and (2) right off, because they don't have portions of the curves which lie in all 4 quadrants (y= x^2, for example, will never have negative y values, will it).

That leaves the circle in (4) or the hyperbola in (3). The circle has the wrong sign of slope in all 4 quadrants, but the hyperbola (as in "Goldilocks and the Three Bears") is j-u-s-t right! So that is the one you want.

Note that for parabolas, hyperbolas, circles, ellipses, and so on displaced from the origin by coordinate changes (y - b = ( x - a )^2 , for example) these regions with certain slope properties will also be displaced.

Now I'll pose you a thought problem. When you consider a parabola of the form y = x^2 , y can never be negative. But that's true only on the real numbers; for complex numbers, y can indeed be negative in that equation, and x is then a complex number (or rather, a pair of them). But to represent a complex number, you need a Cartesian plane, that is, both the original x and y axes. So a function then represents a point-to-point mapping from one Cartesian plane into another Cartesian plane. Could you say that a function has a slope, in that case, and what would that represent?

-- Cheers, -- Mr. d.