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Problem: Given the DE: y^(6) + y = 0, find the general solution.

Solution. I found the roots to be: (sqrt3 - i)/2, (sqrt3 + i)/2, i, -i, (-sqrt3 + i)/2, (-sqrt3 - i)/2

Thus, my general solution led to:

y = c1cost + c2sint + c3exp(t/2*(sqrt3))*cos(t/2) + c4exp(t/2*(sqrt3))*sin(t/2) + c5exp(t/2*-(sqrt3))*cos(t/2) + c6exp(t/2*(-sqrt3))*sin(t/2)

I don't see any errors in this, but please let me know if there are.

When I looked at the solution, the answer is:

y = exp(t/2*(sqrt3))*(c1cos(t/2) + c2sin(t/2)) + c3cost + c4sint*exp(t/2*-(sqrt3))*(c5cos(t/2) + c6sin(t/2)

How exactly did they derive the solution? Is my original answer wrong? The solutions in my original answer seem linearly dependent, and I don't quite see how they derived the solution they did...any help is great!

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# General solution to a 6th order DE

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