Trouble of finding a Polynomial expression

Question

I am trying to find a simple Polynomial expression (u) that follows these conditions:

$$u(0)=u'(1)=0$$

This function must be a degree 2 or 3 polynomial. I want to determine it's unknown coefficients using the Galerkin method. Any suggestions?

Answer 1

There are infinitely many degree $2$ polynomials that satisfy your conditions. In particular,

$$u(x)=ax^2+bx+c$$

satisfies the conditions if and only if

$$u(0)=c=0$$ and $$u'(1) = 2a + b = 0$$

which means all polynomials of the form $ax^2 - 2ax$ for any value of $a$ satisfy your condition.

Furthermore, with degree $3$, you can see that any polynomial of the form $ax^3+bx^2+(-3a-2b)x$ satisfies your condition.