Question

# Solve the differential equation: y''+6y'+12y=0

Second order linear equations
Solve the differential equation:
$$y''+6y'+12y=0$$

2021-02-07
Given the Differential Equation is
$$y''+6y'+12y=0$$
This is a second-order linear differential equation with constant coefficients
Auxilary Equation is
$$m^2+6m+12=0$$
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
$$y=e^{-3x}[C_1\cos\sqrt{3x}+C_2\sin\sqrt{3x}]$$