If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation $y=e^{\left(\frac{dy}{dx}+\frac{d^2y}{dx^2}\right)}$,then the value of $\alpha+\alpha^\beta+\alpha^{2\beta}+\ldots+\alpha^{2023\beta}$ is:

Order of the differential equation of the family of all concentric circles centered at $(h, k)$ is

Assertion $(A)$: The degree of the differential equation $y'' + 2xy' + \log_e\left(\frac{dy}{dx}\right) = 0$ is $2$.
Reason $(R)$: The degree of a differential equation is the highest power of the highest order derivative occurring in the equation, after the equation is expressed in the form of a polynomial in differential coefficients.
The correct option among the following is:

The order and degree of the differential equation $\left(1+\frac{dy}{dx}\right)^{\frac{1}{3}}=\sqrt{\frac{d^2y}{dx^2}}$ are respectively.

Statement $1$: The degrees of the differential equations $\frac{dy}{dx} + y^2 = x$ and $\frac{d^2y}{dx^2} + y = \sin x$ are equal.
Statement $2$: The degree of a differential equation,when it is a polynomial equation in derivatives,is the highest positive integral power of the highest order derivative involved in the differential equation; otherwise,the degree is not defined.

If $m$ and $n$ are the degree and order of the differential equation $\left(1+y_{1}^{2}\right)^{2 / 3}=y_{2}$,then the value of $\frac{m+n}{m-n}$ is