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In general how do we deal with linear second order differential equations with variable coeffecients?
Yes, particularly "Euler type" or "equi-potential" equations.Is general recipe ,but sometimes such eqs can be solved explicitely and in finite ,closed form.
It depends on [itex]\alpha(t),\beta(t)[/itex] functions coefficients involved.
Strictly speaking, "Frobenius" method only applies to series expansion about regular singular points, not general series expansions.The brute force method, usually a method of last resort is the method of frobenius. The problem is you'll generate infinite series solutions which rarely have a closed form. The method is necessary for laplaces equation in cylindrical and spherical coordinates.