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DanJS

  • one year ago

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

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  1. DanJS
    • one year ago
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    general soln

  2. Loser66
    • one year ago
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    characteristic equation is??

  3. DanJS
    • one year ago
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    variation of parameters

  4. Loser66
    • one year ago
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    nope, solve it by characteristic equation.

  5. DanJS
    • one year ago
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    For the homogeneous solution to L(y) = 0 characteristic eq is.. r^2 + 1 = 0 r = + or - i \[y _{h} = c _{1}e^{i * x} + c _{2}e^{-i * x}\]

  6. DanJS
    • one year ago
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    been a few years, bear with me, looking at the book. lol

  7. Loser66
    • one year ago
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    is it not that if \(r=\pm i\) , then \(y_h= C_1 cos (x) + C_2 sin(x)\)??

  8. DanJS
    • one year ago
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    so you can say...from Euler

  9. DanJS
    • one year ago
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    right

  10. Loser66
    • one year ago
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    ok, now solve for partial part.

  11. DanJS
    • one year ago
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    power series expansion of e^(ix)

  12. DanJS
    • one year ago
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    Particular solution to L(y) = f(x), ... hmm

  13. DanJS
    • one year ago
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    minute

  14. DanJS
    • one year ago
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    looking at variation of parameter technique

  15. DanJS
    • one year ago
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    \[y _{p} = g _{1}u _{1} + g _{2}u _{2}\] so, u1 = cos(x) u2 = sin(x)

  16. DanJS
    • one year ago
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    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 ?

  17. DanJS
    • one year ago
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    then integrate to get the g1 and g2 for the particular solution...?

  18. DanJS
    • one year ago
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    i remember memorizing those 2 lines , forgot where they come from, the = 0 and = f(x) , things

  19. Loser66
    • one year ago
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    I am sorry, I am not familiar with this method. I use Wronskian to solve it. :) @ganeshie8 Please.

  20. DanJS
    • one year ago
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    ah right, i did not do that one

  21. DanJS
    • one year ago
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    so i got \[g _{2} = \ln(\sin(x)) ~~~so~~~~g _{1} = -x\] then, \[y _{p} = -x*\cos(x) + \ln(\sin x)*\sin(x)\]

  22. DanJS
    • one year ago
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    u get that with your method?

  23. DanJS
    • one year ago
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    it has to be right, it cancelled nicely, and is a book 'nice' problem.

  24. DanJS
    • one year ago
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    \[y _{g} = y _{h} + y _{p}\]

  25. ganeshie8
    • one year ago
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    looks good, wolfram agrees :) http://www.wolframalpha.com/input/?i=solve+y%27%27+%2B+y+%3D+csc%28x%29++++++++++

  26. DanJS
    • one year ago
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    cool, trying to do a couple of each solution technique to review, i forget a F ton

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