- #1
kq6up
- 368
- 13
I am working on a diff eq. that the prof. did as an example in class.
##y^{\prime\prime}-3y^{\prime}+2y=x^2+x+3##
after subbing in I get:
##2a_2-6a_2x-3a_1+2a_2x^2+2a_1x+2a_o=x^2+x+3##
She set aside the ##x^2## terms and set them equal to zero like such:
##2a_2x^2-x^2=0##
I imagine this ok because each order of exponent can be treated like a orthogonal vector in function space? That is why it is ok to pull these out of the original equation? If I had two vectors that I say have to be equal to zero (these for example):
##\mathbf{A}+\mathbf{B}=0##, so ##a_x\hat{x}+a_y\hat{y}+b_x\hat{x}+b_y\hat{y}=0 ## can be simplified to ##a_x=-b_x## and ##a_y=-b_y##.
Are these two examples analogous? If so, I am satisfied that it works. (not saying it doesn't -- I just want to understand things on a deep level).
Thanks,
Chris
##y^{\prime\prime}-3y^{\prime}+2y=x^2+x+3##
after subbing in I get:
##2a_2-6a_2x-3a_1+2a_2x^2+2a_1x+2a_o=x^2+x+3##
She set aside the ##x^2## terms and set them equal to zero like such:
##2a_2x^2-x^2=0##
I imagine this ok because each order of exponent can be treated like a orthogonal vector in function space? That is why it is ok to pull these out of the original equation? If I had two vectors that I say have to be equal to zero (these for example):
##\mathbf{A}+\mathbf{B}=0##, so ##a_x\hat{x}+a_y\hat{y}+b_x\hat{x}+b_y\hat{y}=0 ## can be simplified to ##a_x=-b_x## and ##a_y=-b_y##.
Are these two examples analogous? If so, I am satisfied that it works. (not saying it doesn't -- I just want to understand things on a deep level).
Thanks,
Chris