PDE with solution in $H^1$

Question

We have $\Omega = (0,1)$ and $\int_{\Omega} e^x u' v' = \int_{\Omega} fv $

$\forall v \in H^1_0(\Omega)$

What is $u \in H^1_0(\Omega)$ if $f=1$?

Answer 1

Using integration by part to get $$\int_0^1 ({\left( {{e^x}u(x)} \right)} '- 1)v(x)dx = 0$$ for all $v$ in $H_0^1(0,1)$, therefore you just have to solve $$(e^{x}u'(x))'=1$$ with Dirichlet boundary condition