Given Dirichlet boundary value problem
$$ \begin{array}[t]{rcl} -\Delta u&=f &\text{ in }\Omega\\ u&=0 &\text{ on }\partial\Omega, \end{array} $$
we can use Green's theorem and transform it into variational form:
$$l(v):=\int_\Omega fv\text{ dx}=\int_\Omega -\Delta u f\text{ dx}=\int_\Omega \nabla u\cdot\nabla v \text{ dx}=: a(u,v),$$
so that the problem boils down to find $u\in H^1(\Omega)$ (Sobolev space) such that:
$$a(u,v)=l(v) \,\,\,\,\forall v \in H^1(\Omega).$$
But now I am given a task that goes in opposite direction:
Which boundary value problem is solved by the problem in variational form:
$$ \begin{array}[t]{rcl} a(u,v)&=&\int_0^1 x^2u'(x)v'(x)\text{ dx}\\ l(v)&=&\int_0^1 v(x)\text{ dx}. \end{array}$$
So we got different $a$ with $x^2$ on the right hand side and this must be some another common PDE, and I can't see which.
Thank you in advance.