Question

Given the linear characteristic polynomial of a differential equation, find the order.
(L+3)^3(L^2-2L+3)=0

Answer 1

order refers to what, the highest derivative?

Answer 2

@amistre64 yes, so 5

Answer 3

5 looks agreeable to me as well

Answer 4

For the linear characteristic polynomial \((r+3)^3 (r^2-2r+3) \)

Answer 5

@amistre64 How do I find the general solution from it?

Answer 6

solve for L

L+3 = 0

L^2 -2L + 3= 0

depending on what the L represents ... its either going to be a string of\[e^{Lx}~or~x^{L}\]

Answer 7

@amistre64 L stands for lambda.

Answer 8

\[(L+3)^3(L^2-2L+3)\]

L^3+3L^2+3L+1
L^2-2L+3
-----------------
\(L^5+3L^4+3L^3+L^2\)
\(-2L^4-6L^3-6L^2-2L\)
\(3L^3+9L^2+9L+3\)
------------------------------
\(L^5+L^4+0L^3+4L^2+7L+3\)

this could come from a setup:
\[y^{(5)}+y^{(4)}+4y''+7y'+3y=0\]

Answer 9

@amistre64 Thank you a lot, you're a lifesaver!

Answer 10

the solution for this can come in a variety of forms:

e^(Lx) is a common evaluation for y

x^(L) is another common evaluation

say y = x^m

y' = m x^(m-1)
y'' = m(m-1) x^(m-2)
y''' = m(m-1)(m-2) x^(m-3)

etc ...

filling in the y''' parts and stripping out the x^{m-5} parts i believe leaves us with a similar characteristic equation

Answer 11

lets assume e^(Lx) is our general solution format

L=-3,-3,-3,1-i sqrt(2), 1+isqrt(2)

when we have multiple roots such as those -3, we have to include some xs to ensure a linearly independant solution of all possible configuration:
\[y=c_1e^{-3x}+c_2xe^{-3x}+c_3x^2e^{-3x}+c_4e^{1-i~\sqrt{2}}+c_5e^{1+i~\sqrt{2}}\]
i believe those i parts can simplify to a cos sin setup