Existence of solution as a power series on non-empty interval Consider the following differential equation:

$$ x''(t) = a_1(t)\, x'(t)+a_0(t)\, x(t) $$

With $a_0,a_1$ functions defined for every real number and continuous on $\mathbb{R}$.

Question: Does there exist a solution which can be written as a converging power series in a non-empty open interval $I$?

To be honest I don't even know the answer yet to the question. It seems maybe unlikely in some sense that this is true because there are several functions that are smooth but not analytic.
This is directly from "Elementary Differential Equations, by Rainville and Bedient", who cites the rigorous proof from "Rainville's Intermediate Differential Equation":

If $ a_1(t),  a_2(t) $ are rational functions of x with denominators that do not vanish at 0

$$x''(t) = a_1(t)y'(t) + a_2(t)y(t)$$
$$x'''(t) = a_1'(t)x'(t) + a_1(t)x''(t) + a'_2(t)x'(t) + a_2(t)x(t) $$

So, $x'''(0)$ can be computed once $x''(0)$ is known.
Using $x'''(t)$, $x''''(t)$ can be computed.  With $x''''(t)$ known, $x''''(t)$ can be computed.

The above process can be continued so long as we wish (with conditions on the derivatives of $ a_1(t),  a_2(t) $, of course).  Therefore, a power series exists based upon Maclaurin's formula, and the initial values $x(0), x'(0)$, is given as:

$$ x(t) = x(0) + x'(0) + \sum_{n=2}^\infty x^{(n)}(0)\frac{x^n}{n!}$$

Where the x' values are computed as in above.
This is directly from "Elementary Differential Equations, by Rainville and Bedient", who cites the rigorous proof from "Rainville's Intermediate Differential Equation":

If $ a_1(t),  a_2(t) $ are rational functions of x with denominators that do not vanish at 0

$$x''(t) = a_1(t)y'(t) + a_2(t)y(t)$$
$$x'''(t) = a_1'(t)x'(t) + a_1(t)x''(t) + a'_2(t)x'(t) + a_2(t)x(t) $$

So, $x'''(0)$ can be computed once $x''(0)$ is known.
Using $x'''(t)$, $x''''(t)$ can be computed.  With $x''''(t)$ known, $x''''(t)$ can be computed.

The above process can be continued so long as we wish (with conditions on the derivatives of $ a_1(t),  a_2(t) $, of course).  Therefore, a power series exists based upon Maclaurin's formula, and the initial values $x(0), x'(0)$, is given as:

$$ x(t) = x(0) + x'(0) + \sum_{n=2}^\infty x^{(n)}(0)\frac{x^n}{n!}$$

Where the x' values are computed as in above.