MHB Second-order homogeneous linear differential equation

[shamieh]

Consider the second-order homogeneous linear differential equation $y'' + 4y' + Ky = 0$

Find the general solution if $K = 4$

So here is what I have:

$r^2 + 4r + 4 = 0 $
=$(r + 2)(r+2)$
$r=-2$ ?

But I thought that you can't do this because you won't be learning anything new if you have two of the same solutions. I'm not sure what to do

 

[MarkFL]

If you have a repeated root $r$, then the general solution is:

$$y(x)=c_1e^{rx}+c_2xe^{rx}$$

 

[shamieh]

so $y = C_1e^{-2t} + C_2 te^{-2t}$

 

[MarkFL]

Yes...as an exercise you may wish to use the reduction of order method to prove the form of the general solution is as I gave above in the case of a repeated root for the characteristic equation. :D