I have to prove that the following bilinear form is not coercive in $H_0^1(0,1)$. $$a(u,v)=\int_0^1x^2u'(x)v'(x)dx$$
Is it as simple as saying since there is no lower bound given on $x^2$ that is positive so we cannot prove that it is coercive?
Also I am trying to find the associated differential equation. For that I tried to test with two different equations and to come up with given the bilinear form. Those are
$$-u''-x^2u=f$$ and $$-xu''=f$$ both with Neumann boundary conditions equal to zero. But cannot seem to get to the bilinear form.