So my question is
In a spherical shell with $1<r<2$ construct a purely radial harmonic function v such that it takes the values $5$ and $4$ at $r=1$ and $r=2$ , respectively
I know that I should use the Laplacian spherical coordinates
and take $Δv=0$
The spherical coordinates
$$\Delta v = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial v}{\partial r}\right) + \frac{1}{r^2\sin^2 \psi}\frac{\partial^2 v}{\partial \theta^2} + \frac{1}{r^2\sin\psi}\frac{\partial}{\partial \psi}\left(\sin\psi \frac{\partial v}{\partial \psi}\right).$$
when I set derivatives in angular directions equal to zero I get
$$ 0 = \frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right). $$
I took the integral and got the solution of $v(r)=$$c_2$- $\frac{c_1}{r}\$ $
But not sure how to proceed from here and Am I solving it the correct way?
Can someone help me with this