differential equation as taylor series

Question

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a series of taylor for $ |t |\ll 1$, $$x(t) = x(0) + \frac{\mathrm{d} x(0)}{\mathrm{d}t} t + \frac{1}{2} \frac{\mathrm{d}^2 x(0)}{\mathrm{d}t^2} t^2 + ...$$

How can i write the solution $x(t)$ in terms of $g(x_0)$, the derivatives $g^{(k)}(x_0)$ and the powers of $t$ , please help.

Answer 1

You have

$\begin{align} x^\prime(t) & = g(x(t)) \\ x^{\prime\prime}(t) & = g^\prime(x(t))x^\prime(t) \\ x^{(3)}(t) & = g^{\prime\prime}(x(t))\left(x^\prime(t)\right)^2 + g^{\prime}(x(t))x^{\prime\prime}(t) \\ ... \end{align}$

Now you have to find a scheme after continuing the derivation of the above equations and you have to set $t=0$ in the end.

Note: It will just make sense, if $x(t)$ is analytic, i.e. the Taylor series has positive radius of convergence.