There has been an example problem, where I asked you to show “that if and are both solutions of the time-dependent Schrödinger equation, than is a solution as well.” Let me review this problem
where in the last line I have use the sum rule for derivatives. This is called the superposition of solutions, and holds for any two solutions to the same Schrödinger equation!
Question: Why doesn’t it work for the time-independent Schrödinger equation?