Essential Boundary Condition and Natural Boundary Conditions in Weak Form Galerkin

Question

Some days ago my teacher gave me a question about using weak form Galerkin to solve an ODE. I'm not so good at solving Differential Equations so I'm here asking for some help(By the way my English is also poor so sorry everybody).
Assuming there's an ODE: $$\begin{align}y''''-10y''+4y+1&=0 \qquad (0<x<2)\tag{1a}\\y(2)&=0\tag{1b}\\y'(0)&=-2\tag{1c}\\y''(2)&=0\tag{1d}\\y'''(0)&=-10\tag{1e}\end{align}$$ I already know the $1st$ and $2nd$ B.C. are both Essential B.C. while the others are Natural B.C., and then according to the weak form Galerkin, I multiply a test function $v(x)$ to the first differential equation and integrate both sides from $0$ to $2$ at the same time. So I get: $$\begin{equation}\begin{aligned}0&=\int_{0}^{2}(y''''-10y''+4y+1)\mathrm{d}x\\&=\int_{0}^{2}\left(y''v''+10y'v'+4yv+v\right)\mathrm{d}x+(y'''-10y')v\big|_{0}^{2}-y''v'\big|_{0}^{2}\\ &=\int_{0}^{2}\left(y''v''+10y'v'+4yv+v\right)\mathrm{d}x-10v(0)\end{aligned}\end{equation}\tag{2}$$ The boundary terms can be simplified to the final form because we know that the value of the test function is $0$ at points with Essential B.C. This is obtained from the relationship between the first-order variation of the functional corresponding to the partial differential equation and the test function.
Next, if we want to use the Weak Form Galerkin to solve this differential equation, following the steps of the Galerkin method, we should assume an approximate function $\widetilde y$ and substitute it into eq.$(2)$. Then, we can use mathematical software such as Mathematica to solve the coefficients and find the approximate solution. However, here comes the problem which bothers me so much:
When assuming the approximate function y, I don't know the minimum order of differentiability required for y, i.e.,if I assume: $$\widetilde y=a_{0}+a_{1}x+a_{2}x^2+\cdot\cdot\cdot+a_{n}x^n\tag{3}$$ If I can get a regular solution from this equation, what should be the minimum value of $n$? In other words, what is the minimum power of x required? Or, how many coefficients are needed at least?
When I tried by myself in Mathematica, I found the minimum value of n should be $2$. It seems this is as a matter of course since the highest derivative order of the term inside the integral in equation $(2)$ is second order. But the question is, the article I found about why the lowest degree of x is quadratic all tells me like this: "The lowest degree only needs to be $2$ because the 'Weak Form' only need to satisfy Essential Boundary Conditions!!!!!" Is this correct? I don't think so, but weak form only satisfy Essential B.C. seems right. And these words really really confuse me.
I wonder, Why the weak form, or the weak form Galerkin only need to satisfy the Essential Boundary Conditions, and seems don't need to satisfy the Natural ones. Actually I've tried to find the reason by Google and Youtube but those reference materials are too focused on physics so I cannot understand. Maybe there's someone can help me with this? I sincerely appreciate any help or reference materials provided by all of you. And I apologize again for those potential communication issues caused by my poor English. If there's some words or equations that I did't express well, please tell me about that and I'll try again. Thanks for help.

Answer 1

The correct weak form is $$0=\int^2_0(y''v''+10y'v'+4yv+v)dx+(y'''−10y')v\Big|^2_0−y''v'\Big|^2_0$$ for all $v$ sufficiently smooth/differentiable with $v(0)=0$ and $v'(2)=0$. Using the provided BC on $y$ and the above restrictions on $v$, this leads to $$0=\int^2_0(y''v''+10y'v'+4yv+v)dx+10v(0)$$

The minimum $n$ is 2 but piecewise, that is you can pick piecewise polynomials of degree 2 both for $y$ and $v$. The essential boundary conditions are directly enforced on the approximation (so that Equations (1b) and (1c) are satisfied by the approximation) while the natural boundary conditions are satisfied in a weak sense only using the above identity (so that (1a), (1d) and (1e) are satisfied in some (weak) sense by the approximation). You could decide to enforce the natural boundary conditions in a strong sense as well by enforcing them directly on the chosen approximation but this is not classical. Probably because of some theorems on existence and uniqueness considerations as well as issues related to convergence analysis but this should be double-checked.