Differential equation degree doubt

Solution 1:

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 $\infty$.

Solution 2:

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.