Quantum mechanics-help!

Homework Statement

Below is a past paper question from a paper i am working through in preperation for my exams. I am having particular trouble with this question, it doesnt seem as straight forward and formulated as other questions. I am unsure if anything i have done is right so any help would be great.

Homework Equations

(a)

Time independent equation

(b)
Time dependent Schrodinger

(c)
Probabilty Density $$=\left|\psi(x,t)\right|^{2}$$

The Attempt at a Solution

a)

$$H_{osc}\phi_{0}(x)=E\phi_{0}(x)$$

$$\\\\H_{osc}\phi_{1}(x)=E\phi_{1}(x)$$

b)

I tried simply putting the wavefunction into the Time dependent Schrodinger equation and differentiating. This mustnt be the way since there is an m on the LHS.

c)

Probabilty Density

$$\left|\psi(x,t)\right|^{2}=1/2[\phi^{2}_{0}+\phi^{2}_{1}+\phi^{*}_{0}\phi_{1}exp(-i\omega t)+\phi^{*}_{1}\phi_{0}exp(i\omega t)]$$

d)

$$\int^{\infty}_{0}\left|\psi(x,t)\right|^{2}dx=1/2\int^{\infty}_{0}[\phi^{2}_{0}+\phi^{2}_{1}+\phi^{*}_{0}\phi_{1}exp(-i\omega t)+\phi^{*}_{1}\phi_{0}exp(i\omega t)]dx$$

^ This doesnt work out too well