Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a power series. Prove its convergence $\forall t$.
How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?
$$ y(t) = \sum_{n=0}^{\infty}a_n t^n $$ thus
$$ \begin{align} y' &=&\sum_{n=1}^{\infty}a_n n t^{n-1}\\ y'' &=&\sum_{n=2}^{\infty}a_n n(n-1) t^{n-2} = \sum_{n=0}^{\infty}a_{n+2} (n+2)(n+1) t^{n} \end{align} $$ thus your equation is
$$ \sum_{n=0}^{\infty}a_{n+2} (n+2)(n+1)t^n + \sum_{n=1}^{\infty}a_n n t^{n-1}\left(\sum_{k=0}^{k=m_1}b_kt^k\right)+ \sum_{n=0}^{\infty}a_n t^n \left(\sum_{k=0}^{k=m_2}c_kt^k\right) = 0 $$ but depending on the forms of $f(t), g(t)$ will determine how to proceed in terms of achieving the recursion relation for $a_n$ which we feed back into $$ y(t) = \sum_{n=0}^{\infty}a_nt^n $$ Here I use $m_1$ and $m_2$ as the degree of the polynomials and assuming that the powers are all $>0$. with the appropriate boundary conditions.