Question

Why is the first assumption in the variation of parameters method valid? The homogeneous solution is:
Yh = C1Y1 + C2Y2 where C1,C2 are constants.

Then we want to find V1 and V2 where the particular solution is:
Yp = V1Y1 + V2Y2, where V1 and V2 are functions.

Yp' = V1'Y1 + V1Y1' + V2'Y2 + V2Y2'

and the assumption I am asking about is that:
V1'Y1 + V2'Y2 = 0.

I have compared the analogous equation for the homogeneous equation:
C1'Y1 + C2'Y2 = 0 and this makes sense to me as the derivatives of C1 and C2 are the derivatives of constants and so equal zero, but it doesn't explain why this assumption is valid for the particular solution. Grateful for any feedback/references.

Answer 1

http://en.wikipedia.org/wiki/Variation_of_parameters

Answer 2

Quoting from the link left by OOOPS:
"We desire A=A(x) and B=B(x) to be of the form A'(x)u_1(x)+B'(x)u_2(x)=0 "

Does not answer the question as to *WHY* this is a valid assumption.

Answer 3

@dan815