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.