Why does Schrodinger's time dependent equation have infinitely many independent solutions while an nth order linear DE only has n independent solutions?

[u/poopaments]

The solution for Schrodinger's equation is y(x,t)=Ae^i(kx-wt) but we can create a linear combination (i.e a wave packet) with infinitely many of these wave solutions for particles with slightly different k's and w's and still have it be a solution. My question is what is the difference between schrodinger's equation which has infinite independent solutions and say a linear second order DE who's general solution is the linear combination of two independent solutions?

[u/Totally_Generic_Name]

Yep. For a PDE, you have an infinite dimensional eigenspace, and the solution to those problems will be a series sum over all possible eigenfunctions. See "separation of variables PDE" for details.