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.
Best Answer
The indicial roots are $0$, $-1$, $-2$. There is a series solution of the form $\sum_{k=0}^\infty c_{2k} x^{2k}$ with $c_0 = 1$, $c_2 = -a/8 - 1/24$, and
$$ -(n+5) b c_n + ((a+1)n + 5a + 3) c_{n+2} + + (n+4)(n+5)(n+6) c_{n+4} = 0 $$
and a series solution of the form $\sum_{k=0}^\infty c_{2k-1} x^{2k-1}$ with $c_{-1}=1$, $c_1 = -a/3$, and this same recurrence. A third fundamental solution involves $x^n$ for even $n \ge -2$ and $x^n \ln(x)$ for even $n \ge 0$.