Final answer:
A function is a solution to a differential equation if it correctly substitutes into the equation for all values of the independent variable and satisfies any given initial conditions, ensuring the function's behavior aligns with the resultant equation.
Step-by-step explanation:
A function is considered a solution to a differential equation if it satisfies certain criteria. The most definitive criterion is that it satisfies the differential equation for all values of the independent variable. This means that when substituting the function and its derivatives into the differential equation, the left-hand side and right-hand side of the equation are equal. Additionally, if there are initial or boundary conditions specified, the function must satisfy these as well.
In a broader context, if a function is continuous and differentiable, this ensures the function and its derivatives exist and can be plugged into the differential equation. Being a linear combination of other solutions is related to the superposition principle, which applies to linear differential equations. Here the functions that make up the solution set can be combined to form new solutions.
Thus, while it is essential for a function to satisfy the differential equation for all values of the independent variable, additional conditions such as initial conditions must also be met for it to be considered a complete solution. It is important that the function follows the behavior dictated by the equation across all relevant points in the domain.