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•• We have seen that second-order differential equations like the Schrödinger equation have two independent solutions. Consider the fourth-order equation d4ψ/dx4 = β4ψ, where β is a positive constant. Prove that each of the four functions e±βx and e±iβx is a solution. (It is a fact — though harder to prove — that any solution of the equation can be expressed as a linear combination of these four solutions.)
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