Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 37A / i / Solution

For $\Psi =f(R){\sin }^2\theta$, the angular differentiation givesTrying $f=R^m$ in the fourth-order equation gives the indicial productwhose four distinct roots are $-1,1,2,4$. This proves that the stated four-term expression is the general separated solution of this angular type.

Put $z=b/a$ and $F(s)=A/s+Bs+C{s}^2+D{s}^4$. The velocity formulas become $u_R=2UF(s)\cos \theta /s^2$ and $u_{\theta }=-U{F}^{\prime }(s)\sin \theta /s$. The four boundary conditions therefore give