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

Question
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
$$m^2+6m+12=0$$
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
$$y=e^{-3x}[C_1\cos\sqrt{3x}+C_2\sin\sqrt{3x}]$$

### Relevant Questions

$$y=3e^{3x}$$ is a solution of a second order linear homogeneous differential equation with constant coefficients. The equation is:
(a) $$y''-(3+a)y'+3ay=0$$, a any real number.
(b) $$y''+y'-6y=0$$
(c) $$y''+3y'=0$$
(d) $$y''+(3-a)y'+3ay=0$$, a any real number.
(e) Cannot be determined.
Solve for the general solution of the given special second-ordered differential equation
$$xy''+x(y')^2-y'=0$$
Find the general solution of the second order non-homogeneous linear equation:
$$y''-7y'+12y=10\sin t+12t+5$$
Solve the following non-homogeneous second order linear differential equation.
$$y''+py'-qy=-rt$$
if $$p=4,\ q=8,\ r=7$$.
Solve the second order linear differential equation using method of undetermined coefficients
$$3y''+2y'-y=x^2+1$$
A dynamic system is represented by a second order linear differential equation.
$$2\frac{d^2x}{dt^2}+5\frac{dx}{dt}-3x=0$$
The initial conditions are given as:
when $$t=0,\ x=4$$ and $$\frac{dx}{dt}=9$$
Solve the differential equation and obtain the output of the system x(t) as afunction of t.
Let $$y_1$$ and $$y_2$$ be solution of a second order homogeneous linear differential equation $$y''+p(x)y'+q(x)=0$$, in R. Suppose that $$y_1(x)+y_2(x)=e^{-x}$$,
$$W[y_1(x),y_2(x)]=e^x$$, where $$W[y_1,y_2]$$ is the Wro
ian of $$y_1$$ and $$y_2$$.
Find p(x), q(x) and the general form of $$y_1$$ and $$y_2$$.
One solution of the differential equation $$y" – 4y = 0$$ is $$y = e^{2x} Find a second linearly independent solution using reduction of order. asked 2020-11-02 Solve the Differential equations \(2y′′ + 3y′ − 2y = 14x^{2} − 4x − 11, y(0) = 0, y′(0) = 0$$
$$4x^2-3y=0,\ y(1)=3,\ y'(1)=2.5,$$ the basis of solution are $$y_1=x^{-\frac{1}{2}}$$ and $$y_2=x(\frac{3}{2})$$