Question

The differential equation dy/dx=x+1/y-3

I. produces a slope field with vertical tangents at y = 3
II. produces a slope field with vertical tangents at x = -1
III. produces a slope field with rows of parallel segments

Answer 1

I only
II only
III only
I and III

Answer 2

If you think back to learning y=mx+b in middle school algebra class the slope is m, and they say rise over run because it's the ratio of \(m=\frac{\Delta y}{\Delta x}\). So a single line has the same slope everywhere, but this is a differential equations so instead of the slope being the same line everywhere, the slope is different at each point in the xy plane.

In order to solve this, don't let this weird \(\frac{dy}{dx}\) confuse you, you can replace it by \(m\) just for the time being, and imagine that at each point (x,y) the slope is a line right there.

\[m = x+\frac{1}{y}-3\]

So if I look in the xy plane at the point (2,-1) then the slope is \(m=2+\tfrac{1}{-1}-3 = -2\) so if it was a line, imagine the line \(y=-2x\) going through the point (2,-1). I can plot the slope at that point, then you try plotting some more points. That's how you create a slope field.



I made the point red and the slope blue. When you fill in the rest you can get an understanding of WHAT a slope field is so that you can see how to answer the question.