I am curious if anyone has heard of this problem before:
Suppose that $u(x,p)$ is a function of $x$ and $p$. These arguments need not be scalars.
Let $u(x,p)$ satisfy some differential equation, say: $\frac{d^2}{dx^2}u(x,p) + f(x,p)\frac{d}{dx}u(x,p) + c(x,p)=0$. $f$ and $c$ are any functions just to make this equation sensible.
Now if one is interested in $\frac{d}{dp}u(x,p)$, one way is to differentiate the above differential equation with respect to $p$, reorder the derivatives, and solve the differential equation for $\frac{d}{dp}u(x,p)$.
Someone told me that there's another way to do this using the so-called adjoint method. I've been looking around but I am not sure if I have the correct references. Anyone seen this before?