Question

The solution to a differential equation is y=e^x^3. Which of the following direction fields would correspond to that differential equation?

Answer 1

To determine the direction field that corresponds to the differential equation y=e^x^3, we need to find the derivative of y with respect to x.

Using the chain rule, we have:

dy/dx = d/dx (e^x^3) = 3x^2 e^x^3

Now, we can use this derivative to plot the direction field. At each point (x,y), we draw a short line segment with slope equal to dy/dx evaluated at that point.

Out of the given direction fields, the one that corresponds to y=e^x^3 is:

```
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```

This direction field has vertical line segments at each point, which corresponds to the fact that the derivative dy/dx = 3x^2 e^x^3 is always positive.