Dirichlet boundary condition.(Solving differential condition.)

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Here's my problem. It means 'Finding solution by using polar coordinates.

I m trying to solve this so I find these expressionwhat I wrote here is ksi(xi). Sorry for bad writing. And please ignore the right-upper side

I conclude it has result when $n = 2$ only in $cos(x)$. So solution what I found was $2 + \frac{r^2}{8}$

But the solution paper says it is $8+r^2(cos(\theta)-1/2)$ .

I dont get what is wrong.