Final answer:
The student's task is to verify if certain functions are solutions to a given differential equation and to determine if these solutions are part of a fundamental set. This involves mathematical operations such as differentiation and substitution into the original equation, as well as checking for linear independence.
Step-by-step explanation:
The student's question pertains to the verification of whether given functions y1 and y2 are solutions to a specified differential equation, and whether these solutions form a fundamental set. Verification involves substituting the functions into the differential equation and checking if they satisfy the equation. A fundamental set of solutions implies that the functions are linearly independent, and they span the solution space of the differential equation. This is important in fields like Quantum Mechanics where the superposition principle depends on a set of linearly independent solutions to the Schrödinger equation.
To validate the given functions as solutions, we would differentiate them as necessary and plug them back into the given differential equation. If both functions satisfy the equation independently, they are indeed solutions. To determine if they form a fundamental set, we would check their Wronskian or see if one can be expressed as a multiple of the other. If the Wronskian is non-zero or they cannot be expressed as multiples of each other in their domain, they indeed form a fundamental set of solutions.