Given initial values
$$b_0,\left(\frac{db}{da}\right)_0,....,\left(\frac{d^{n-1}b}{d{a^{n-1}}}\right)_0$$
I could approximate the differential equation
$$\frac{d^nb}{d{a}^n}=f\left(b,\frac{db}{da},....,\frac{d^{n-1}b}{d{a^{n-1}}}\right)$$
using the formulas
$$\left(\frac{d^nb}{d{a^n}}\right)_0=f\left(b_0,\left(\frac{db}{da}\right)_0,....,\left(\frac{d^{n-1}b}{d{a^{n-1}}}\right)_0\right)$$
$$b_m\approx{b_{m-1}}+\left(\frac{db}{da}\right)_{m-1}+...+\frac{1}{(n-1)!}\left(\frac{d^{n-1}b}{d{a^{n-1}}}\right)_{m-1}(\Delta{a})^{n-1}+\frac{1}{n!}\left(\frac{d^nb}{d{a^n}}\right)_{m-1}\left(\Delta{a}\right)^n$$
$$\left(\frac{db}{da}\right)_m\approx\left(\frac{db}{da}\right)_{m-1}+\left(\frac{d^2b}{d{a^2}}\right)_{m-1}(\Delta{a})+...+\frac{1}{(n-2)!}\left(\frac{d^{n-1}b}{d{a^{n-1}}}\right)_{m-1}(\Delta{a})^{n-2}+\frac{1}{(n-1)!}\left(\frac{d^nb}{d{a^n}}\right)_{m-1}\left(\Delta{a}\right)^{n-1}$$
$$.$$
$$.$$
$$.$$
$$\left(\frac{d^{n-1}}{d{a^{n-1}}}\right)_m\approx\left(\frac{d^{n-1}b}{d{a^{n-1}}}\right)_{m-1}+\left(\frac{d^nb}{d{a^n}}\right)_{m-1}\Delta{a}$$
$$\left(\frac{d^nb}{d{a^n}}\right)_m\approx{f\left(b_m,\left(\frac{db}{da}\right)_m,....,\left(\frac{d^{n-1}b}{d{a^{n-1}}}\right)_m\right)}$$
with $m$ being a positive integer
I understand that one way to approximate a differential equation is to use Euler's Method, and from what I understand about Euler's Method this method is the same as Euler's Method for a first order differential equation, but I'm not sure if this would be considered a variation of Euler's Method for higher order differential equations.
Is there a name for this method of approximating a differential equation?