This exercise deal with the equation $\Delta u=0$ in polar coordinates. Suppose the inner side of the annulus $D=\{(r,\theta)\,:\,r_{0}\leq r\leq 1\,;\, 0\leq\theta<2\pi \}$ is insulated and the outer side is held at temperature $u(1,\theta)=f(\theta)$.
My problem is to interpretate this informations. I think "inner side insulated" means $u_{\theta}(r_{0},\theta)=0$. So, I only have the boundary conditions: $u_{\theta}(r_{0},\theta)=0$ and $u(1,\theta)=f(\theta)$. Don't I need two more boundary conditions?
There are only two boundaries in an annulus. Visually you have
insulated for the heat conduction problem means its a Neumann problem
$$\frac{\partial u}{\partial t} = k\frac{\partial^{2}u}{\partial x^{2}} \tag{1} $$
$$ \frac{\partial u}{\partial x}(0,t) = 0 \tag{2}$$ $$ \frac{\partial u}{\partial x}(L,t) = 0 \tag{3}$$
for your problem if you solve it in annulus
$$\frac{\partial u}{\partial r}(r_{0},\theta) = 0 \\ u(1,\theta) = f(\theta) \tag{4} $$
so you're supposed to solve
$$\frac{d^{2}\phi}{dr^{2}} = -\lambda r \tag{5} $$
with those boundary conditions.