I have a function like follows,
$\psi(x) = A\,\psi_1(x) + B \,\psi_2(x)$
I have a full numerical solution for $\psi(x)$, I want to provide an analytical solution to $\psi(x)$ by finding suitable constants $A$ and $B$ to match analytical functions $\psi_1(x)$ and $\psi_2(x)$ to the numerical solution.
What is the easiest way to do this?
Perhaps the simplest way of doing this would just be to use least-squares or some other simple data fitting method; since you have a numerical solution for $\psi(x)$ then you only have a finite number of points $\{x_i\}$ for which you know $\psi(x)$, then just fit $$ \min_{A, B} \sum_i \left(\psi(x_i) - A\psi_1(x_i) - B\psi_2(x_i)\right)^2. $$
Additionally, if there are particular points you care about, you can reweight those in the objective to minimize their loss instead by some weights $w_i\ge 0$ (e.g. if you care that points $x_3, x_4$ are very close to correct, say, because they're near some asymptote you care about, then make $w_3, w_4$ large):
$$ \min_{A, B} \sum_i w_i\left(\psi(x_i) - A\psi_1(x_i) - B\psi_2(x_i)\right)^2. $$
Note that both of these problems have analytic solutions and many packages which can do the fitting for you (see, e.g., scikit-learn).