We are given:
$\tag 1 \displaystyle \frac{d^2u}{dr^2} +\frac{2}{r} \frac{du}{dr} = 0,~~ R_1 \le r \le R_2, ~~u(R_1) = V_1, ~~u(R_2) = 0$
If $R_1 = 2, R_2 = 4$, and $V_1 = 110$ volts, then how can we approximate $u(3)$ using the Linear Shooting Method.
Well, we would like to compare our numerical results to the exact results, so the exact result will be given by:
$\tag 2 \displaystyle u(r) = \frac{V_1~R_1}{r}\left(\frac{R_2-r}{R_2-R_1}\right)$
Evaluating $(2)$ at $r = 3$ for the given lengths and potential in $(1)$, we arrive at:
$\tag 3 \displaystyle u(3) = \frac{100 ~\times ~ 2}{3}\left(\frac{4 - 3}{4 - 2}\right) = \frac{110 ~ \times ~2}{3} \frac{1}{2} = \frac{110}{3} \approx 36.66667.$
From $(3)$, you know what the Linear Shooting Method should produce at the end!
For the algorithm, we want to setup the linearized functions, :
$\tag 4 \displaystyle \frac{d^2u}{dr^2} = - \frac{2}{r} \frac{du}{dr}$
We have $\displaystyle p(r) = - \frac{2}{r}, q(r) = 0$ and $r(r) = 0$.
From $(1)$, we have:
$h = \displaystyle \frac{b-a}{N} = \frac{R_2 - R_1}{N} = \frac{4 - 2}{N} = \frac{2}{N}.$ (They have must have given you an N.)
That should be enough to get you going because you now have everything needed for the setup and the final answer you should be getting - you only need crank out the Runge-Kutta numbers.
Note: I know you are saying you need to do this by hand which is fine, however, in practice, you want to code it up so once you have a setup, you can crank out the numbers without making silly algebra mistakes and to check your work. You need not submit your working code, but are using it to learn and derive results in an efficient manner.
