Consider the differential equation

y$7$y$+10$y$=$cos$(3$t$).$y$7$y$+10$y$=$cos$(3$t$).$

$($a$)$ Find r$1$r$1,$ r$2$r$2,$ roots of the characteristic polynomial of the equation above.

r$1,$r$2=$r$1,$r$2=$

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$($b$)$ Find a set of real$-$valued fundamental solutions to the homogeneous differential equation corresponding to the one above.

y$1($t$)=$y$1($t$)=$

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y$2($t$)=$y$2($t$)=$

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$($c$)$ Find a particular solution ypyp of the differential equation above.

yp$($t$)=$yp$($t$)=$