I am trying to solve this third order differential equation with non-constant coefficients $$(x^3)y'''+6 x^2 y''+[6+(1+a-bx^2)x^2]x y'+[1+3 a-5 b x^2]x^2 y=0$$ where $a$ and $b$ are constants and $y$ is a function of $x$ only.
The problem originated form a Micropolar fluid flow problem involving longitudinal and torsional oscillations. The original differential equation was 4th order and a combination of the Laplace transform, a change in variables as well as multiple integral transforms were used to bring the form given below.
I have tried the standard methods for solving differential equations with variable coefficients, as well as the Frobenius power series method, using x =0 as a regular singular point, however it became too complicated for a pattern and hence solutions to be formed.
I would be grateful if anyone had any further ideas on how to solve this differential equation.