The schroedinger equation for harmonic oscillator can be written:
E*psi = [(h^2)/2m][((d^2)*psi)/(dx^2)] + (1/2)kx^(2*psi)
Write and formally differentiate each term to get the second derivative with respect to X. Put it all into the equation as shown and you will see that there will be an infinite number of possible solutions, with progressively higher powers of x in the wave function. Write the first three (lowest three in order of powers of X) wave functions that solve the equation (that is, determining values for the A's in each of the possible solutions).
We solve and find Hermite's differential equation. Its solution is Hermite polynomials (H_n)(y), (u_n)(y) = (a_n)(H_n)(y). This allows us to further solve and differentiate.