DE question: (2+m)^2 = 0

Question

adamaero · Answer

So m_1 = -2 = m_2. But when I'm finding a differential equation to the auxillary equation, why is the "x" in there:
y = c_1*e^(-2x) + c_2*x*e^(-2x) + ...

zepdrix · Answer

Why is the general solution in the form of exponentials?
Is that what you're asking?

adamaero · Answer

nope

zepdrix · Answer

Oh oh the repeated solution! I see :)

zepdrix · Answer

Hmm good question :3 I'm trying to remember why that is..

adamaero · Answer

I thought it was c_1*e^... (only)

adamaero · Answer

but the solution has a c_2*x*e^...

adamaero · Answer

attachment

zepdrix · Answer

http://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx The top half of the page explains where that extra x is coming from with repeated roots. It's a little lengthy so I can't really comment on it. But the process makes sense.

unklerhaukus · Answer

$$(2+m)^2=0\
m=-2,-2\
y(x) = (A+Bx)e^{-2x}$$

unklerhaukus · Answer

repeated root ==> resonance

adamaero · Answer

https://www.youtube.com/watch?v=HlJbHMLaORk&feature=youtu.be&t=80 https://youtu.be/hV8xVnb_EmY?t=144 Meh, I will just believe =)