# Differential Equations Assignment (small step needs clarification)

When given

y'' + ay' + by + c = 0

The characteristic equation is:

λ^2 + aλ + b = 0

Now I'm making a substitution of e^(λx) to get

[(λ^2 + aλ + b)][e^(λx)] = 0

double root : [(λ - alpha)^2] [e^(λx)] = 0

My question now is that if I were to do the same thing with the following equation would it follow that

y''' + ay'' + by' + cy +d = 0

Characteristic equation λ^3 + aλ^2 + bλ + c

then making the same substitution ( λ^3 + aλ^2 + bλ + c ) e^(λx)

NOW HERE IS MY QUESTION: Can I write the above as

[(λ - alpha)^3][e^(λx)]

or do I have to write it a different way. Thank you very much.. I know this doesn't really have much to do with the solving of differential equations but it would be very helpful if I knew that that was correct for my assignment.

Thanks.

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rock.freak667
Homework Helper
NOW HERE IS MY QUESTION: Can I write the above as

[(λ - alpha)^3][e^(λx)]

or do I have to write it a different way. Thank you very much.. I know this doesn't really have much to do with the solving of differential equations but it would be very helpful if I knew that that was correct for my assignment.

Thanks.
You can write it like that, if the roots are repeated.