Solve: y'' + y = csc(x) ; 0

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Solve: y'' + y = csc(x) ; 0

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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general soln
characteristic equation is??
variation of parameters

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nope, solve it by characteristic equation.
For the homogeneous solution to L(y) = 0 characteristic eq is.. r^2 + 1 = 0 r = + or - i \[y _{h} = c _{1}e^{i * x} + c _{2}e^{-i * x}\]
been a few years, bear with me, looking at the book. lol
is it not that if \(r=\pm i\) , then \(y_h= C_1 cos (x) + C_2 sin(x)\)??
so you can say...from Euler
right
ok, now solve for partial part.
power series expansion of e^(ix)
Particular solution to L(y) = f(x), ... hmm
minute
looking at variation of parameter technique
\[y _{p} = g _{1}u _{1} + g _{2}u _{2}\] so, u1 = cos(x) u2 = sin(x)
and \[g _{1}^{~'}*u _{1} + g _{2}^{~'}*u _{2} = 0\] \[g_{1}^{'}*u _{1}^{'} + g_{2}^{'}*u _{2}^{'} = f(x)\] so just put in u1 and u2, and solve g1 and g2 prime ?
then integrate to get the g1 and g2 for the particular solution...?
i remember memorizing those 2 lines , forgot where they come from, the = 0 and = f(x) , things
I am sorry, I am not familiar with this method. I use Wronskian to solve it. :) @ganeshie8 Please.
ah right, i did not do that one
so i got \[g _{2} = \ln(\sin(x)) ~~~so~~~~g _{1} = -x\] then, \[y _{p} = -x*\cos(x) + \ln(\sin x)*\sin(x)\]
u get that with your method?
it has to be right, it cancelled nicely, and is a book 'nice' problem.
\[y _{g} = y _{h} + y _{p}\]
looks good, wolfram agrees :) http://www.wolframalpha.com/input/?i=solve+y%27%27+%2B+y+%3D+csc%28x%29++++++++++
cool, trying to do a couple of each solution technique to review, i forget a F ton

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