Question: Recall the method for solving an nth-order homogenous differential equation by finding roots of an auxiliary equation.any (n) an -1y (n -1)... asy aoy =0Rewrite

Recall the method for solving an nth-order homogenous differential equation by finding roots of an auxiliary equation.any (n) an -1y (n -1)... asy aoy =0Rewrite this as a polynomial in m, replacing the jth derivative with m'.anm" an -1m"-1... a,m ao =0Then for every root m, of multiplicity k of this polynomial, the general solution for the equation must contain the following linear combination.We are given the following third-order homogeneous differential equation.y'"12y"36y'=0Therefore, the auxiliary equation is a third-degree polynomial, which can be factored as follows.m312m236m =0m(m212m 36)=0m(m 6)(m (C73)=0inter an exact numberSolving for m, the roots of the auxiliary equation are m1=with multiplicity 1, and m2=with multiplicity 2.

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