Q:

Boundary conditions of ODE are derivatives, constant falls out

I am trying to solve an ordinary differential equation, and I have the fundamental solution, but do not know how to get the constants.
The solution is $ T(r)=Ar^2+C_1\ln(r)+C_2$,
I have the conditions $\left.\frac{dT}{dr}\right|_{r=r_i}=B$ and $\left.\frac{dT}{dr}\right|_{r=r_o}=D$
where $r_i$ and $r_o$ are the inner and outer radius. 
I took $\frac{dT}{dr}= 2Ar+\frac{C_1}{r}$ with $r=r_i$ to find $C_1$ but have no idea how to find $C_2$.
My thought was to integrate as $\int\frac{dT}{dr}dr=\int D dr =Dr = T(r)$ then evaluate at $r = r_o$ but do not think this is valid.

A:

Since both boundary conditions are on the derivative, and $C_2$ is an additive constant, which therefore does not appear in the derivative, you cannot determine $C_2$ from the data.
(But you get two conditions for $C_1$, and if they are incompatible, there is no solution at all – unless you also get to choose $A$.)

