The auxiliary equation for the differential equation $$ x^{2} y^{\prime \prime}+3 x y^{\prime}+y=6 $$ is Select the correct answer. $$ m^{2}+2 m+1=0 $$ $$ m^{2}+3 m+1=6 $$ $$ m^{2}+3 m+1 $$ $$ m^{2}+3 m+1=0 $$ $$ m^{2}+2 m+1=6 $$

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We start by considering a trial solution of the form  
$$
y=x^m.
$$  
Then, the derivatives are  
$$
y' = m x^{m-1}, \quad y'' = m(m-1)x^{m-2}.
$$

Substitute these into the differential equation  
$$
x^{2} y'' + 3 x y' + y = 6.
$$

The left-hand side becomes  
$$
x^2\left[m(m-1)x^{m-2}\right] + 3x\left[m x^{m-1}\right] + x^m = m(m-1)x^m + 3mx^m + x^m.
$$

Factor out $$ x^m $$:  
$$
\left[m(m-1) + 3m + 1\right]x^m = 0.
$$

For nontrivial solutions (where $$ x^m \neq 0 $$), the coefficient must vanish:  
$$
m(m-1) + 3m + 1 = 0.
$$

Simplify the expression:  
$$
m^2 - m + 3m + 1 = m^2 + 2m + 1 = 0.
$$

Thus, the auxiliary equation is  
$$
m^2 + 2m + 1 = 0.
$$
The auxiliary equation is $$ m^{2} + 2m + 1 = 0 $$.

The auxiliary equation for the differential equation is
Select the correct answer.

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The auxiliary equation for the differential equation is Select the correct answer.