OK, let's remember a few things...

solving the time-independent Schrodinger equation leads immediately to the time dependent solution, so we focus on the time-independent one
solving the time independent equation:

$\hat{H} ϕ = E ϕ$

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actually means solving it for every $E$ such that a solution exists.
It is therefore a eigenvalue-problem.
Furthermore, since our $\hat{H}$ is nice and self-adjoint (like all observables), it also forms a basis of the space. This basis is made of infinitely many basis functions.
therefore, just like we did when solving solving partial differential equations with the Fourier series, solving the problem basically means finding the eigenvector basis, and then expressing the initial condition in terms of it.
Due to linearity, we can just add each component up, and it all works out.

Now, there are two ways to go about this.

The first is the stupid "here's a guess", and in this case, there is an explicit basis that can be proven complete that actually works: Hermite functions.

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The second is the much celebrated ladder operator method.

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