How to solve this differential equation for $\psi_n$?:

$$\frac{1}{2}\frac{\partial^2}{\partial x^2}\psi_n=\lambda_n\psi_n$$

apparently this is a heat equation but I cannot find information on this. Any help is much appreciated. thanks.


The boundary conditions are for initial and terminal, respectively, $\psi_n(x_0)=0$, and $\psi_n(x_T)=0$.

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    $\begingroup$ What are your boundary conditions? Where did this equation come from? What was the starting problem? Please add more context $\endgroup$ – Dylan Feb 25 at 5:36
  • $\begingroup$ @Dylan, the boundary conditions are 0 at both the initial and terminal points $x_0$ and $x_T$...and I'm still trying to see how to solve for $c_1$ and $c_2$....and the lambdas. I really want to take the $ln$ of both sides, but obviously that will not work when there is a zero. Is a taylor expansion useful here? $\endgroup$ – nundo Feb 25 at 5:40

No, this is not "a heat equation", but solving it (with appropriate boundary conditions) is typically one of the steps in solving a heat equation boundary value problem using separation of variables. The general solution of your equation is $\psi_n = c_1 \exp(\sqrt{2\lambda_n} x) + c_2 \exp(-\sqrt{2\lambda_n} x)$.


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