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RolyPoly
 3 years ago
Solve \(y^{(4)}  8y''' + 26y''  40y' + 25y =0 \)
RolyPoly
 3 years ago
Solve \(y^{(4)}  8y''' + 26y''  40y' + 25y =0 \)

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RolyPoly
 3 years ago
Best ResponseYou've already chosen the best response.0\[y^{(4)}  8y''' + 26y''  40y' + 25y =0 \] \[λ^4  8λ^3 + 26λ^2 40λ + 25 =0 \] \[(λ^2 4λ + 5)^2 = 0\] λ = (2+i) (repeated) or λ = 2i (repeated) \[y = c_{1} e^{(2+i)x} + c_{2} x e^{(2+i)x} + c_{3} e^{(2i)x} + c_{4} xe^{(2i)x} \]\[= e^{2x} (c_{1} e^{ix} + c_{2} x e^{ix} + c_{3} e^{ix} + c_{4} xe^{ix})\]\[= e^{2x} [c_{1} e^{ix} + c_{3} e^{ix} + x(c_{2} e^{ix} + c_{4} e^{ix})]\]\[= e^{2x} [c_{1} (cosx + isinx) + c_{3} (cos(x)+i(sin(x)) \]\[\ \ \ \ \ \ + x(c_{2} (cosx + isinx) + c_{4} (cos(x)+i(sin(x)))]\]\[= e^{2x} [(Acosx + Bsinx) + x(Ccosx + Dsinx) \]

RolyPoly
 3 years ago
Best ResponseYou've already chosen the best response.0The last line should be \[= e^{2x} [(Acosx + Bsinx) + x(Ccosx + Dsinx)]\]

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0so far ... seems correct!!
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