Solving the boundary value problem by means of Galerkin method

I have a task which should be solved with Galerkin method: $$y''-0.5x^2y+2y=x^2 \\ y(1.6)+0.7y'(1.6)=2 (1)\\ y(1.9)=0.8 (2)$$ I already solved it with other methods so the correct answer I know, but I can't solve it exactly with Galerkin method.

The task was programmed in Mathematica. For the simple boundary values such as: $$y(a)=L\\ y(b)=Q$$ where L and Q - digits, a and b - boundaries of range.

$\phi_k$ for the simple boundary values has appearance: $\phi_k(x)=(x-a)(x-b)^k$

so the answer we can get like: $y(x)=\phi_0(x)+\sum_{k=0}^n a_k\phi_k(x)$

The main problem is finding $\phi_k$ for boundary values such as (1) and (2). Maybe someone already faced the same problem.

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What is the meaning of $2(1)$ and $0.8(2)$ ? Are they typo? – doraemonpaul Sep 14 '12 at 2:58
@doraemonpaul, It just the references are used in the last sentence (I meant digits (1) and (2)). – Roman Shkarin Sep 14 '12 at 3:09