So I want to determine the coefficients regarding the BDF($2$). Note that I'm absolute beginner, so my attempt is not that great.
The solution has to be: $ \frac{3}{2}y_{k+2} - 2y_{k+1} + \frac{1}{2}y_{k} = hf(t_{k+2},y_{k+2})$
My work:
so first we insert $m=2$ into the definition:
So we want the lagrange interpolation polynomial $ q \in \mathbb{P^2} $ to the points $ (t_{k},y_{k}),(t_{k+1},y_{k+1}) $ and $ (t_{k+2},y_{k+2}) $, where $y_{k+2}$ is the solution of the following equation: $q'(t_{k+2}) = f(t_{k+2},y_{k+2}) $.
Next I read an article about this topic and try so reconstruct it, so:
starting with the Lagrange basis and note that $ \lambda_j(t_{k+i}) = \delta_{ji} $.
So we have $ q(t_{k+i}) = \sum_{j=0}^2 \lambda_j(t_{k+i})y_{k+j} = y_{k+i}$.
It follows $q (t) = \sum_{j=0}^2 \lambda_j(t)y_{k+j} $.
Now with the condition $ q'(t_{k+2}) = f(t_{k+2},y_{k+2}) $ we get:
$ \sum_{j=0}^2 \alpha_j y_{k+j} = f(t_{k+2},y_{k+2}) $, where $ \alpha_j = \lambda^{'}_{j}(t_{k+2}) $.
Here is the point, where I'm stuck. I know that we have to determine the $ \alpha_{j} $, but honestly I don't know how.
I hope you can help me out, thank you in advance.
Now you look up what the Lagrange kernel functions actually look like, $$ λ_0(t)=\frac{(t-t_1)(t-t_2)}{(t_0-t_1)(t_0-t_2)} \implies λ_0'(t)=\frac{2t-(t_1+t_2)}{(t_0-t_1)(t_0-t_2)} $$ and similar for the other functions.