I've recently been introduced to orthogonality of Legendre polynomials, I think I understand the idea behind it; that the integral of the products of two polynomials in the range $(1,-1)$ will equal $0$.
My issue is that I'm finding it quite confusing to apply this idea. I've been given some questions to look at and think about, I'll only put one of them so that I can take the ideas and apply them to the others myself.
Here's what's troubling me with each question/what I think I might be able to do:
a) This question is probably really straightforward but to be honest I'm not even sure what they're asking me to write, do I need to state the definition I mentioned above?
b) This question is troubling me quite a bit. I believe we have $\int_{-1}^1 P_1(x)(x^2+bx+c)dx=0$ and $\int_{-1}^1P_0(x)(x^2+bx+c)= 0$ since they are orthogonal, but I'm not sure where to go from here or how to use these to obtain $b$ and $c$, could I equate them to each other or use substitution to obtain them?
c) I'm more looking to sharpen my understanding of orthogonality so this question isn't a massive concern, saying that, it's always better to get as much help as I can so if you could show me how to do this I would appreciate it, we haven't quite got to this yet in the course.