# Paul's question at Yahoo! Answers regarding a 3rd order linear homogeneous ODE

#### MarkFL

Staff member
Here is the question:

Differential equations factoring?

Find the general solution to the following

y'''-8y=0
I have posted a link there to this topic so the OP can see my work.

#### MarkFL

Staff member
Hello Paul,

We are given to solve:

$$\displaystyle y'''-8y=0$$

The associated characteristic equation is:

$$\displaystyle r^3-8=(r-2)(r^2+2r+4)=0$$

Hence, the roots are:

$$\displaystyle r=2,\,-1\pm i\sqrt{3}$$

and so the solution is:

$$\displaystyle y(x)=c_1e^{2x}+e^{-x}(c_2\cos(\sqrt{3}x)+c_3\sin(\sqrt{3}x))$$