Schrodinger's equation (explanation to non physicist)

For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the explanation. However I'm not so stupid to think I can understand this landmark without some years of physics. I'll just be happy to be able to read the equation and recognize it in its various forms.

To be more precise, here are my questions.

Hyperphysics tell me that Shrodinger's equation "is a wave equation in terms of the wavefunction".

1. Where is the wave equation in the most general form of the equation?

$\mathrm{i}\hslash \frac{\mathrm{\partial}}{\mathrm{\partial}t}\mathrm{\Psi}=H\mathrm{\Psi}$

I thought wave equation should be of the type

$\frac{{\mathrm{\partial}}^{2}}{{\mathrm{\partial}}^{2}t}u={c}^{2}{\mathrm{\nabla}}^{2}u$

It's the difference in order of of derivation that is bugging me.

2. Can somebody show me the passages in a simple (or better general) case?

3.I think this questions is the most difficult to answer to a newbie. What is the Hamiltonian of a state? How much, generally speaking, does the Hamiltonian have to do do with the energy of a state?

4.What assumptions did Schrödinger make about the wave function of a state, to be able to write the equation? Or what are the important things I should note in a wave function that are basilar to proof the equation? With both questions I mean, what are the passages between de Broglie (yes there are these waves) and Schrödinger (the wave function is characterized by)?

5.It's often said "The equation helps finds the form of the wave function" as often as "The equation helps us predict the evolution of a wave function" Which of the two? When one, when the other?

For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the explanation. However I'm not so stupid to think I can understand this landmark without some years of physics. I'll just be happy to be able to read the equation and recognize it in its various forms.

To be more precise, here are my questions.

Hyperphysics tell me that Shrodinger's equation "is a wave equation in terms of the wavefunction".

1. Where is the wave equation in the most general form of the equation?

$\mathrm{i}\hslash \frac{\mathrm{\partial}}{\mathrm{\partial}t}\mathrm{\Psi}=H\mathrm{\Psi}$

I thought wave equation should be of the type

$\frac{{\mathrm{\partial}}^{2}}{{\mathrm{\partial}}^{2}t}u={c}^{2}{\mathrm{\nabla}}^{2}u$

It's the difference in order of of derivation that is bugging me.

2. Can somebody show me the passages in a simple (or better general) case?

3.I think this questions is the most difficult to answer to a newbie. What is the Hamiltonian of a state? How much, generally speaking, does the Hamiltonian have to do do with the energy of a state?

4.What assumptions did Schrödinger make about the wave function of a state, to be able to write the equation? Or what are the important things I should note in a wave function that are basilar to proof the equation? With both questions I mean, what are the passages between de Broglie (yes there are these waves) and Schrödinger (the wave function is characterized by)?

5.It's often said "The equation helps finds the form of the wave function" as often as "The equation helps us predict the evolution of a wave function" Which of the two? When one, when the other?