Schrödinger equation can be understood as a linear, second order, partial differential equation
There exists other interpretations of this equation as a first order linear equation in time for an element on a Hilbert Space . In this context, you can define a linear operator called Hamiltonian
And re interpret the Schrödinger equation as
It has great interest to discuss the case , this is, time independent potentials. Then a particular family of solutions can be "easily" found by imposing that
As in the usual examples of separation of variables, we will found that a separation constant is needed, call itE.. that can be interpreted as the energy of the system. Among other things it will happen that
And a very interesting feature (depending on the potential ) is that there may be more than one separation constant, it may be happen that there exists even an infinite denumerable set that solves the equation, so you just can't drop some of them, and you will have to take the sum or even series.
Now, to the cat. Suppose that you're given the Schrödinger cat potetial and that the most general solution is
This is the solution. The system is neither alive nor dead, it is in both systems at the same time. Now, another axiom of Quantum Mechanics is that when you measure the quantity deadness or liveness of the cat it will be either alive or death, not both of course. But after measuring the state is a linear superposition of both states. So the Schrödinger cat is a very poor choice of words to replace linear superposition. Which of course doesn't happen in classical mechanics, and this is why it is wierd. But I'm afraid that Nature is quantum-mechanical, strange for us or not.