Final answer:
The given differential equation has a unique solution passing through a point (x0, y0) if the function (1 + y³) is continuously differentiable in a region around the point.
Step-by-step explanation:
To determine a region of the xy-plane for which the given differential equation (1 + y³)y' = x² would have a unique solution passing through a point (x0, y0), we need to consider the existence and uniqueness theorem for ordinary differential equations. This theorem guarantees the existence and uniqueness of a solution when certain conditions are met. In this case, the conditions are satisfied if the function (1 + y³) is continuously differentiable in a region around the point (x0, y0).
Applying this condition, we can see that the region of the xy-plane for which the differential equation has a unique solution passing through the point (x0, y0) is where the function (1 + y³) is continuously differentiable.