Order of Differential Equation containing integration term?

Question

What is the order and degree of the following equation? $$\Big(\frac{d^2y}{dx^2}\Big)^3 + \frac{dy}{dx} = \int y dx.$$

Answer 1

It really depends. When you have an integration term, it's not longer an ODE, thus it doesn't have the same classification.

If you want to turn it into an ODE, you'd have to differentiate, thus rasing the order to $3$:

$$3 (y'')^2 y'''+y''=y$$

Answer 2

Set $u=\int y\,dx$. Then the ODE form of your mixed integral-differential equation is $$ (u''')^3+u''=u $$ which has order $3$ and is polynomial with degree $3$.

Note that in the given form this is not an ODE and thus it makes no sense to speak of a degree.