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In this YouTube video about the particle in a box problem:

He determines that $\psi(x)=A\sin(kx)+B\cos(kx)$ is the solution to the time-independent Schrödinger equation, which is: $$\frac{\partial^2\psi}{\partial x^2}+k^2\psi=0$$ but he decides to not explain how this solution was determined. I was hoping somebody would be able to help me understand how to come up with that equation on my own. I know this is probably very simple, but I’m trying to learn quantum mechanics while not being very knowledgeable in math.

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  • $\begingroup$ The equation is saying that the second derivative of a function is a constant multiple of the function itself. The only elementary functions that satisfy this property are sine and cosine (which are representations of complex exponentials). $\endgroup$
    – Vincent Thacker
    Commented Sep 11, 2021 at 6:46

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A lot of differential equations are not solved by a fixed method, but rather by guessing and playing around with the equation until you find an insight. So, since there's not a formal 'derivation' for guessing, people often just give you the solution. Yours is an example of a differential question that is often just guessed.

How to guess it? Notice that your differential equation says that two derivatives just give a constant factor in front of the original function. You might remember that the second derivatives of sines and cosines just give back sines and cosines, up to a factor. So, playing around with sines and cosines until you find the most general working solution would solve this equation for you.

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