Differential Equation

Question

The differential equation  .

I. will have a slope field with negative slopes in quadrant I

II. will have a slope field with positive slopes in all quadrants

III. will produce a slope field with columns of parallel tangents

Question options:

1)	I only
2)	II only
3)	III only
4)	None of these

Answer 1

A slope field is a visual representation of a differential equation of the form dy/dx = f(x, y). Since the derivative is the slope of the original equation, at each sample point (x, y), you can plug the x and y coordinates into the differential equation to find the slope of the original function at that point. By mapping out the slope field, you can get a general idea of what the original equation can look like without having to solve the for the original function.

dy/dx = (x/y)²

At all points with x=0, dy/dx = (0/y)² = 0 so the slope will be a horizontal line.

At all points with y=0, dy/dx = (x/0)² = undefined so the slope will be a vertical line.

At all other points, dy/dx = (x/y)² = positive (since x/y is squared) so the slope will be positive.

Option 2 is correct since the slope will be positive in all quadrants.