How are eigenfunctions found? Let's stick to the derivative operator and any other operators built up from it. For instance, if $D = \frac{d}{dt}$, then the eigenfunction of $D$ is known to be $e^{at}$, associated with eigenvalue $a$. However, what about for a more complicated operator? Even something as simple as $1 + D$. Is there a general method that works for any such operator? For that matter, how was the eigenfunction for $D$ itself found?

For a minimal example an eigenfunction to the operator ${\bf T = D}^2+2{\bf D}$ on the space of $\\{\sin(x),\cos(x)\\}$.

The operator $D$ on this space is a matrix: $ {\bf D} = \begin{bmatrix}0&1\\\\-1&0\end{bmatrix}$

Therefore $${\bf T} = {{\bf D}^2+2{\bf D}} = \left[\begin{array}{cc}-1&2\\\\-2&-1\end{array}\right]$$

We can now solve $\det({\bf T}-\lambda {\bf I})$

$\lambda = -1\pm 2i$ $e = [\pm \sqrt{1/2} i , \sqrt{2}]^T$

So if we allow complex coefficients, apparently second derivative plus 2 times derivative of $$\left(\frac{\partial^2}{\partial x^2} +2 \frac{\partial}{\partial x}\right) \\{i\sin(x)+\cos(x)\\} = (-1+2i)(i\sin(x)+\cos(x))$$

And we can check it on Wolfram Alpha\(i*sin\(x\)%2Bcos\(x\)\)%20%2B%20\(d%2Fdx\)%202*\(i*sin\(x\)%2Bcos\(x\)\)\)%2F\(-1%2B2i\)).