Mathematics

Problem

Let

\displaystyle \frac{dy}{dx}=\frac{x+1}{y-1}

with domain restricted to points where

y\ne 1

. a. Explain how the magnitude and sign of

\dfrac{dy}{dx}

behave as

y

approaches 1 from above and from below. Include a discussion of the limiting behavior of the slopes. b. On axes covering

-2\le x\le 2

and

-2\le y\le 4

, sketch a representative slope field for the differential equation. Then draw and clearly label the solution curves passing through the points

(-1,0)

and

(0,2)

, showing any asymptotic behavior that occurs. c. Give a rigorous argument to show that no solution curve of the differential equation can cross the horizontal line

y=1

. Your reasoning should rely solely on properties of the differential equation, not on any explicit solution formula.

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