Given the Differential Equation is

\(y''+6y'+12y=0\)

This is a second-order linear differential equation with constant coefficients

Auxilary Equation is

\(m^2+6m+12=0\)

Solve This quadratic equation.

Compare with:

\(ax^2+bx+c=0\)

So roots are

\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

Here,

\(a=1,\ b=6,\ c=12\)

Hence,

\(m=\frac{-6\pm\sqrt{36-48}}{2}\)

\(=\frac{-6+-\sqrt{-12}}{2}\)

\(=\frac{-6+-\sqrt{3i}}{2}\)

\(=-3\pm\sqrt{3i}\)

So here Roots of equations are complex

So Solution of Ordinary differential equation with root a+ib and a-ib is

\(y=e^{ax}[C_1\cos{bx}+C_2\sin{bx}]\)

Here \(a=-3\) and \(b=\sqrt3\)

So the solution is

\(y=e^{-3x}[C_1\cos\sqrt{3x}+C_2\sin\sqrt{3x}]\)

\(y''+6y'+12y=0\)

This is a second-order linear differential equation with constant coefficients

Auxilary Equation is

\(m^2+6m+12=0\)

Solve This quadratic equation.

Compare with:

\(ax^2+bx+c=0\)

So roots are

\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

Here,

\(a=1,\ b=6,\ c=12\)

Hence,

\(m=\frac{-6\pm\sqrt{36-48}}{2}\)

\(=\frac{-6+-\sqrt{-12}}{2}\)

\(=\frac{-6+-\sqrt{3i}}{2}\)

\(=-3\pm\sqrt{3i}\)

So here Roots of equations are complex

So Solution of Ordinary differential equation with root a+ib and a-ib is

\(y=e^{ax}[C_1\cos{bx}+C_2\sin{bx}]\)

Here \(a=-3\) and \(b=\sqrt3\)

So the solution is

\(y=e^{-3x}[C_1\cos\sqrt{3x}+C_2\sin\sqrt{3x}]\)