StellarTo verify that a function is a solution to a differential equation, we substitute the function into the equation and check whether the left side and right side are equal. This is different from solving a differential equation from scratch. Here, the function is already given, and the job is to confirm that it satisfies the derivative relationship. This matters because a differential equation may have many solutions, sometimes infinitely many, and verification helps show whether a specific function belongs to that family.
The process is straightforward: first compute the derivative required by the equation, then plug both the function and its derivative into the differential equation, and finally simplify. If both sides match for all values in the domain, then the function is a solution. If they do not match, then it is not a solution. This idea reinforces that solutions to differential equations are functions, not just numbers, and many different functions can satisfy the same derivative rule.