Direction fields

A direction field is shown above.
Which of the four differential equations below could generate this direction field? \[ \begin{align} \frac{dy}{dx} &= x^2 - y^2 \quad &(A)\\ \frac{dy}{dx} &= \frac{y+xy}{x} \quad &(B)\\ \frac{dy}{dx} &= -\frac{x}{y} \quad &(C)\\ \frac{dy}{dx} &= 2y \quad &(D)\\ \end{align}\]

Describe the features of the direction field

The direction field has:

Vertical tangents along the x axis
Horizontal tangents along the y axis
Tangents close to -1 along the line y = x
Tangents close to 1 along the line y = -x

Eliminate candidate equations

For equation A, y' = 0 when y = x. This eliminates A.
For equation B, when x > 0 and y > 0 then y' > 0. This eliminates B.
For equation C, y' = 0 when x = 0 and y' is vertical when y = 0. So C is a candidate.
For equation D, y' is constant for a given value of y. This eliminates D.
After eliminating equations A, B and D, the only remaining candidate is C.

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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