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$\mathrm{Find}~ u \in H_0^1(\Omega) ~\mathrm{such~that}~ A(u,v)=F(v), \quad \forall v \in H_0^1(\Omega)$ ,

$\mathrm{where}~ A(u,v) = \int_{\Omega} a \nabla u \cdot \nabla v + b u v, \quad \quad F(v) = \int_{\Omega} f v$ .