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We'll start this section off with the material which most text books that will cover in this section. We will take the matter from the Second Order chapter and expand this out to nth order linear differential equations. As we will see almost all of the 2nd order material will very naturally extend out to nth order along with only some bit of new material.
Therefore, let's start things off this time with several fundamental concepts for nth order linear differential equations. The most general nth order linear differential equation is as,
Pn(t) y(n) + Pn-1(t) y(n-1) + ......+ P1(t) y' + P0(t) y = G(t) (4)
So we hopefully recall this,
y(m) = dmy/dxm
Several of the theorems and concepts for this material need that y(n)has a coefficient of 1 and therefore if we divide out by Pn (t) we find,
y(n) + pn-1(t) y(n-1) + ......+ p1(t) y' + p0(t) y = g(t) (5)
As we may suspect an initial value problem for an nth order differential equation will need the subsequent n initial conditions.
y(t0) = y?0, y'(t0) = y?1, ........ y(n-1) (t0) = y? (n-1) = y?(n-1), (6)
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Triple Eigenvalue with 1 Eigenvector The eigenvalue/vector pair in this case are l and ?h , . Since the eigenvalue is real we know as the first solution we require is, e l
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We will be assuming here that our roots are of the form, in this case, r 1,2 = l + mi If we take the first root we'll find the following solution. x l + m i It i
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