Differential equation degree doubt $$\frac{dy}{dx} = \sin^{-1} (y)$$ The above equation is a form of $\frac{dy}{dx} = f(y)$, so degree should be $1$.  But if I write it as $$y = \sin\left(\frac{dy}{dx}\right)$$ then degree is not defined as it is not a polynomial in $\frac{dy}{dx}$. Please explain?
 A: The explanation is simple: they are not the same equations. Even if two equations are equivalent, they are not exactly the same. For example:
$$\frac{dy}{dx}=\sqrt[3]{x}\tag1$$
$$\left(\frac{dy}{dx}\right)^3=x\tag2$$
The equation $(1)$ is not the same as equation $(2)$ even if they do have exactly the same solutions (in $\mathbb R$ to be clear). You can see that $(1)$ has degree $1$ and $(2)$ has degree $3$.
The problem is when you try to find degree of i.e. $$y=e^{y'}
\quad\text{or}\quad
y=\sin\left(\frac{dy}{dx}\right)\tag{a,b}$$
There exist a formula that allow you define a degree of non-polynomials, namely $$\deg\;f(x)=\lim_{x\to\infty}\frac{\log|f(x)|}{\log(x)}$$ but in some cases, such as $(b)$, is unlikely to work, whereas for other cases it allows to define a degree of non-polynomial functions. For example equation $(a)$ may be degree $\infty$.
A: Degree is not defined for terms of the form $\sin(\frac{dy}{dx})$ because, on expanding the sinusoid, the degree of the highest power goes to infinity.
In order to find the degree corresponding to the differential equation, first bring it to its standard form. Try to express the equation as a polynomial function of the derivatives. Then the power corresponding highest order is called the degree of the equation. Hence the degree of the equation you mentioned is 1.
A: To make sense of the definition of the order of a differential equation, you need to first isolate the highest order derivative first. For example, 
$\frac{dy}{dx} = y, \frac{dy}{dx} = y^2 + 1$ are of the first order. 
$ \frac{d^2 y}{dx^2} = y, \frac{d^2 y}{dx^2} = (\frac{dy}{dx})^3 + 1$ are of second order.
Your differential equation $\frac{dy}{dx} = \sin^{-1}y$ is first order even if it can be written in the equivalent form $y = \sin(\frac{dy}{dx}).$
