# Undetermined coefficient in recurrence relation

I am given $$3x^2(x+2)y''+7xy'-2y=0, x \geq 0$$. I am asked to solve this differential equation with a series solution around $$x=0$$. Note, however that $$x=0$$ is a regular singular point since:

$$P(x) = 3x^2(x+2) \implies 3(0)^2(0+2)= 0,$$ $$\lim_{x \to 0}x\frac{Q(x)}{P(x)} = \lim_{x \to 0}x\frac{7x}{3x^2(x+2)} = \frac{7}{6} \text{ and}$$ $$\lim_{x \to 0}x^2\frac{R(x)}{P(x)} = \lim_{x \to 0}x^2\frac{-2}{3x^2(x+2)} = -\frac{1}{3}$$ are both finite. Thus we assume a solution of the form

$$y = \sum_{n=0}^\infty a_n x^{n+r}, y' = \sum_{n=0}^\infty a_n(n+r) x^{n+r-1}, \\y''= \sum_{n=0}^\infty a_n(n+r)(n+r-1) x^{n+r-2}$$

Which, after some algebra, yields the indicial equation $$(2r+1)(3r+5)=0 \implies r_1 = -\frac{1}{2}, r_2 = -\frac{3}{5}$$ And the recurrence relation $$a_n=-\frac{-3a_{n-1}(n-3/2)(n-5/2)}{(6n+1)(n-1)}, \text{for} \ n \geq 1$$ and $$a_0$$ is arbitrary. However, I soon realized that $$a_1$$ is undetermined as the denominator becomes 0. I was wondering if I got my algebra wrong or the entire approach is not correct. Thanks

• I suppose that you made some mistake since you should have $r_1=-\frac 23$ and $r_2=\frac 12$ (hoping no mistake on my side). – Claude Leibovici Sep 25 '18 at 8:03
• This is a differential equation of the second order and I expect that you should be able to pick two constants freely (like $a_0$ and $a_1$), not just one. All other values can be evaluated from the recurrence relation. – Oldboy Sep 25 '18 at 8:52

The collected coefficients for the power $$n+r$$ are $$0=6(n+r)(n+r-1)a_n+3(n+r-1)(n+r-2)a_{n-1}+7(n+r)a_n-2a_n\\ =[6(n+r)^2+(n+r)-2]a_n+3(n+r-1)(n+r-2)a_{n-1}$$ with the convention that $$a_{-1}=0$$. Then the indicial equation for $$n=0$$ is $$0=6r^2+r-2=(2r-1)(3r+2)$$ with recursion $$a_n=-\frac{3(n+r-1)(n+r-2)}{(2n+2r-1)(3n+3r+2)}.$$ The recursion for $$r=\frac12$$ reduces to $$a_n=-\frac{3(2n-1)(2n-3)}{4n(6n+7)}a_{n-1},$$ and for $$r=-\frac23$$ $$a_n=-\frac{(3n-5)(3n-8)}{3n(6n-7)}a_{n-1}.$$ This gives totally problem free recursions for the coefficient sequences of the two basis solutions.