Final answer:
If one solution to a second-order homogeneous ordinary differential equation is even, the other must be b. odd, ensuring that the solutions are linearly independent and can form a basis set.
Step-by-step explanation:
The question pertains to the behavior of solutions to a second-order homogeneous ordinary differential equation with a given parity (odd-even). When faced with homogeneous linear differential equations, the solutions often display relationships based on their parity, which refers to their symmetry with respect to the y-axis. To answer the question: If one solution to a second-order homogeneous ordinary differential equation is even, the other must be odd. This is due to the fact that the even and odd functions form a basis set for the solutions of these equations.
This is a manifestation of the common principle in linear differential equations that different linearly independent solutions can be combined to form the general solution. In this case, an even and an odd function can provide such a basis. Moreover, when considering the parity of products of functions, an even function times an even function remains even, an odd times an odd also produces an even, but an even times an odd gives an odd. Since the solutions should be linearly independent, if one solution is even, the other must be odd to maintain their independence.
The integral property mentioned in the reference, stating that the integral over all space of an odd function is zero, serves as another illustration of the distinct behaviors of even and odd functions. Similarly, the alternating nature of solutions between even and odd functions as indicated in the reference about wave functions reinforces the alternating pattern expected in the solutions of second-order differential equations.