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Find the complementary solution of [itex]y^\left(4\right) + 2y'' + y = sint[/itex]

Homogeneous Form would be [itex]y^\left(4\right) + 2y'' + y = 0[/itex]

[itex]r^4 + 2r^2 + r = 0 \rightarrow r(r^3 + 2r + 1) = 0[/itex]

This is where I'm stuck. Once I find [itex]y_c(t)[/itex] I should be able to finish the problem, but I'm having trouble at this step. What would be the next step here?

The book's solution is [itex]y_c(t) = C_1 cost + C_2 sint + C_3 tcost + C_4 tsint[/itex] which would suggest complex numbers involved here.

Homogeneous Form would be [itex]y^\left(4\right) + 2y'' + y = 0[/itex]

[itex]r^4 + 2r^2 + r = 0 \rightarrow r(r^3 + 2r + 1) = 0[/itex]

This is where I'm stuck. Once I find [itex]y_c(t)[/itex] I should be able to finish the problem, but I'm having trouble at this step. What would be the next step here?

The book's solution is [itex]y_c(t) = C_1 cost + C_2 sint + C_3 tcost + C_4 tsint[/itex] which would suggest complex numbers involved here.

**Edit: Found my error, it was in the r equation.**
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