Final answer:
The question involves determining the regions of the xy-plane where specific differential equations are defined. Though equation (a) is not a differential equation, the rest represent first and second-order differential equations with solutions valid over the entire xy-plane.
Step-by-step explanation:
The student asks about determining regions of the xy-plane for specific differential equations. For equation (a) y = x² - 4x + 4, this is not a differential equation but simply a quadratic equation representing a parabola that opens upwards with vertex (2,0). This function is defined for all x-values on the xy-plane.
For equation (b) y' + 2xy = 0, this is a first-order linear homogeneous differential equation, and solutions can exist across the entire xy-plane. The function y = ce-x², where c is an arbitrary constant, is a family of solutions to this differential equation.
Equation (c) y'' - y = 0 represents a second-order linear homogeneous differential equation. Solutions to this equation are in the form y = c1ex + c2e-x, where c1 and c2 are constants. These solutions are valid for all x in the xy-plane.
Lastly, for equation (d) dy/dx = ex, this is an example of a separable differential equation, and the solution is y = ex + C, where C is the constant of integration, valid across the entire xy-plane.