Solution of Constant Coefficient Linear nth Order ODE
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Proof Technique
Consider the linear second order ODE with constant coefficients:
- $(1): \quad \ds \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$
where $a_k$ is a constant for $0 \le k \le n$ and $\map R x$ is a function of $x$.
The general solution to $(1)$ can be found as follows.
- Find the roots $m_1, m_2, \ldots, m_n$ of the auxiliary equation
- $(2): \quad \ds \sum_{k \mathop = 0}^n a_k m^k = 0$.
If $(2)$ has distinct roots, then the general solution $\map {y_g} x$ is of the form:
- $\ds y_g = \sum_{k \mathop = 0}^n A_k e^{m_k x}$
If there are repeated roots of $(2)$, further needs to be done.
Let $m_j$ be a repeated root of $(2)$ with multiplicity $r$.
Then the left hand side of $(2)$ can be written:
- $\map P m \paren {m - m_j}^r$
where $\map P m$ is a polynomial which does not contain the factor $m - m_j$.
Then the $r$ instances of $m_j$ give rise to the solutions:
- $(4): \quad y = A_{j_0} e^{m_j x} + A_{j_1} x e^{m_j x} + \dotsb + A_{j_{r - 1} } x^{r - 1} e^{m_j x}$
Proof
Let the reduced equation of $(1)$ be written:
- $(3): \quad \paren {a_n D^n + a_{n - 1} D^{n - 1} + \dotsb + a_1 D + a_0} y = 0$
By factoring the left hand side we get:
- $\map P D \paren {D - m_j}^r y = 0$
It remains to be shown that $(4)$ is a solution to $(3)$.
Each of the terms is of the form $p e^{m_j x} u$, where $u$ is a power of $x$ and $p$ is a constant.
So:
\(\ds \map {\paren {D - m_j} } {p e^{m_j x} u}\) | \(=\) | \(\ds \map D {p e^{m_j x} u} - m_j p e^{m_j x} u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p e^{m_j x} D u\) |
\(\ds \map {\paren {D - m_j}^2} {p e^{m_j x} u}\) | \(=\) | \(\ds \map {\paren {D - m_j} } {p e^{m_j x} D u}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p e^{m_j x} \map D {D u}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p e^{m_j x} \map {D^2} u\) |
and so on until:
\(\ds \map {\paren {D - m_j}^r} {p e^{m_j x} u}\) | \(=\) | \(\ds p e^{m_j x} D^r u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
because $D^r u = 0$ when $u = 1, x, x^2, \ldots, x^{r - 1}$.
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Sources
- 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 3$. Equations of higher order and systems of first order equations: $\S 3.1$ The $n$th order equation