10.2: Superposition of time-dependent solutions

There has been an example problem, where I asked you to show "that if $\psi_1(x, t)$ and $\psi_2(x, t)$ are both solutions of the time-dependent Schrödinger equation, than $\psi_1(x, t)+\psi_2(x, t)$ is a solution as well." Let me review this problem

$$\begin{aligned} -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} \psi_1(x, t)+V(x) \psi_1(x, t) & =\frac{\hbar i \partial}{\partial t} \psi_1(x, t) \\[8pt] -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} \psi_2(x, t)+V(x) \psi_2(x, t) & =\frac{\hbar i \partial}{\partial t} \psi_2(x, t) \\[8pt] -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2}\left[\psi_1(x, t)+\psi_2(x, t)\right]+V(x)\left[\psi_1(x, t)+\psi_2(x, t)\right] & =\frac{\hbar i \partial}{\partial t}\left[\psi_1(x, t)+\psi_2(x, t)\right]\end{aligned}$$

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?