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Homework Statement
Solve the IVP
(x^2)y'' + 4xy' - 40y = x^6
for y(1) = 10, y'(1) = 1
Homework Equations
not so much "equations" but here I try to use variation of parameters to get the particular solution.
The Attempt at a Solution
FOR THE HOMOGENEOUS SOLUTION:
using the substitution y = x^r I get a characteristic equation of
r^2 - 3r -40 = 0
so,
(r-8)(r+5) = 0
and then, the homogeneous solution will be
yh = c1|x|^8 + c2|x|^-5
FOR THE PARTICULAR SOLUTION:
Here is where I run into trouble. I'm unsure of how this part is usually done, but I'm guessing that since the original problem does not have constant coefficients, we are ofrced to use variation of parameters (can someone confirm this? It seems right to me but maybe there's a trick I'm missing here).
so, first I get the wronskian, which is the determinant of this 'matrix':
|x|8 |x|-5
8|x|7 -5|x|-6
I get -5|x|2 - 8|x|2 = -13|x|2
then here's my main problem (or maybe my problem is earlier -- not sure if I'm doing this right). I get to the variation of parameters part, and there's two integrals I have to do:
integral of |x|8x6 divided by 13|x|2
and integral of |x|-5x6 divided by 13|x|2
here is where I get confused: how do I deal with the integrals with absolute values? Did I do something completely wrong or is there something I'm not seeing? Is there some way to avoid this mess? Thanks for any help, my textbook never gets into the specifics of how to deal with an euler equation when it isn't homogeneous
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