- #1
Skrew
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I was wondering how the general formula for the solutions of an nth order linear homogeneous ODE that had a characteristic equation which could be factored to (x-a)^n was derived(IE a set of solutions consisting e^(mx), x*e^(mx), ...x^(n-1)e^(mx)))?
For example the ODE,
y^(3)- 3y'' + 3y' - y = 0,
with characteristic equation,
m^3 - 3m^2 + 3m - 1 = 0
can be factored to
(m-1)^3,
where m = 1
and e^(x), x*e^(x) and x^2*e^(x) are all solutions.
For a second order ODE this can be found using a reduction of order technique but for higher order ODE's it gets very difficult to do so I am wondering what proof/explanation exists to show that we know such solutions exist?
I have looked around online and all the books/articles just say that's the case but don't provide an explanation.
For example the ODE,
y^(3)- 3y'' + 3y' - y = 0,
with characteristic equation,
m^3 - 3m^2 + 3m - 1 = 0
can be factored to
(m-1)^3,
where m = 1
and e^(x), x*e^(x) and x^2*e^(x) are all solutions.
For a second order ODE this can be found using a reduction of order technique but for higher order ODE's it gets very difficult to do so I am wondering what proof/explanation exists to show that we know such solutions exist?
I have looked around online and all the books/articles just say that's the case but don't provide an explanation.
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