Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them?

In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-dependent) Schrödinger equation, which reads $$ \Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right]. $$ It is clear that this is not a linear combination of the stationary states that he found previously in the chapter.

Does that mean that solving the time-dependent Schrödinger equation by separation of variables does not yield the general solution as the author claimed? if so, how do we find the other solutions?

Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them?

In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-dependent) Schrödinger equation, which reads $$ \Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right]. $$ It seems that this is not a linear combination of the stationary states that he found previously in the chapter.

If it is the caes, does that mean that solving the time-dependent Schrödinger equation by separation of variables does not yield the general solution as the author claimed? if so, how do we find the other solutions?

Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them  ?

In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-dependent) Schrödinger equation, which reads $$ \Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right] $$ It is clear that this is not a linear combination of the stationary states that he found previously in the chapter.

Does that mean that solving the time-dependent Schrödinger equation by separation of variables does not yield the general solution as the aothor claimed  ? if so, how do we find the other solutions  ?

Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them?

In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-dependent) Schrödinger equation, which reads $$ \Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right]. $$ It is clear that this is not a linear combination of the stationary states that he found previously in the chapter.

Does that mean that solving the time-dependent Schrödinger equation by separation of variables does not yield the general solution as the author claimed? if so, how do we find the other solutions?