Final answer:
To verify solutions to a differential equation, derivatives of the functions are substituted into the equation. The linear independence of functions is determined using the Wronskian, where a non-zero value indicates independence.
Step-by-step explanation:
The student's question pertains to the topic of differential equations and their solutions, as well as the concept of linear independence concerning functions that may satisfy these equations. In order to verify whether functions satisfy a given differential equation, one would typically take the derivatives as required by the equation and substitute them back into the equation to check for correctness. For instance, if we take functions such as e^(x/2) and xe^(x/2) and we're given a differential equation, we would differentiate these functions and substitute to verify if they indeed are solutions.
Linear independence of functions such as e^(x/2) and xe^(x/2) can be determined using the Wronskian, denoted as w(ex/2, xex/2), which involves the calculation of a determinant of a matrix consisting of the functions and their derivatives. If the Wronskian is non-zero, the functions are linearly independent. If the Wronskian is zero, the functions may or may not be dependent; further investigation is required.