We are given the following third$-$order homogeneous differential equation.

$y^{\text{'}\text{'}\text{'}}12y^{\text{'}\text{'}}36y^{\text{'}}=0$

Therefore, the auxiliary equation is a third$-$degree polynomial, which can be factored as follows.

Solving for $m,$ the roots of the auxiliary equation are $m_1=0,0$ with multiplicity $1,$ and $m_2=-6,-6$ with multiplicity $2.$

Step $2$

From the root $m_1=0$ with multiplicity $1,$ we know this means the general solution must contain $c_1{e}^{(0)x}=c_1.$

From the root $m_2=-6$ with multiplicity $2,$ we know this means the general solution must contain $c_2{e}^{-6x}{c}_{3}x{e}^{-6x}.$

Therefore, the general solution of the differential equation contains the sum of these terms.

$y=c_1{c}_2{e}^{-6x}{c}_3()$

$$ SKIP $($YOU CANNOT COME BACK$)$