Hi I try solve the following problem of differential equation
$$ x''+tx'+\frac{1}{1+t+t^2}x=0\tag 1$$ when $$x(1)=0\ \ \ ;\ \ \ x'(1)=1 $$
is the solution analytic in $t_0=1$ and his convergence radius is $R>1$?
Ok, I think I need put the differential equation like a frobenius differential equation, then I get, with the initial equation, that my solution is define by $$ \varphi_1(t)=\sum_{n=0}^{\infty} a_n(t-1)^{n+1}$$ $$\varphi_2(t)=C\varphi_1(t)\log(t-1)+\sum_{n=0}^{\infty} b_n(t-1)^n $$
I can not work with $\varphi_1$ in $(1)$, I do not know... someone could help me?