MHB Justifying the Method of Undetermined Coefficients

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The discussion focuses on justifying the method of undetermined coefficients in solving ordinary differential equations (ODEs) using the annihilator method. It presents a table outlining specific forms of the non-homogeneous term \( g(x) \) and the corresponding particular solution \( y_p(x) \), emphasizing the need for the smallest non-negative integer \( s \) to ensure no terms in \( y_p(x) \) overlap with the homogeneous solution. The conversation also highlights the derivation of an annihilator for a specific function \( g(x) \) and the resulting auxiliary equation. Additionally, participants suggest minor edits for clarity and consistency in notation, particularly regarding operator representation. The discussion concludes with an acknowledgment of improvements made to the original content.
MarkFL
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The annihilator method can be used to derived the entries in the following table for the method of undetermined coefficients, familiar to all students of ordinary differential equations:

[TABLE="class: grid, width: 750, align: center"]
[TR]
[TD]Type[/TD]
[TD]$g(x)$[/TD]
[TD]$y_p(x)$[/TD]
[/TR]
[TR]
[TD](I)[/TD]
[TD]$p_n(x)=a_nx^n+\cdots+a_1x+a_0$[/TD]
[TD]$x^sP_n(x)=x^s\left(A_nx^n+\cdots+A_1x+A_0 \right)$[/TD]
[/TR]
[TR]
[TD](II)[/TD]
[TD]$ae^{\alpha x}$[/TD]
[TD]$x^sAe^{\alpha x}$[/TD]
[/TR]
[TR]
[TD](III)[/TD]
[TD]$a\cos(\beta x)+b\sin(\beta x)$[/TD]
[TD]$x^s\left(A\cos(\beta x)+B\sin(\beta x) \right)$[/TD]
[/TR]
[TR]
[TD](IV)[/TD]
[TD]$p_n(x)e^{\alpha x}$[/TD]
[TD]$x^sP_n(x)e^{\alpha x}$[/TD]
[/TR]
[TR]
[TD](V)[/TD]
[TD]$p_n(x)\cos(\beta x)+q_m\sin(\beta x)$
where $q_m(x)=b_mx^m+\cdots+b_1x+b_0$[/TD]
[TD]$x^s\left(P_N(x)\cos(\beta x)+Q_N(x)\sin(\beta x) \right)$
where $Q_N(x)=B_Nx^N+\cdots+B_1x+B_0$ and $N=\max(n,m)$[/TD]
[/TR]
[TR]
[TD](VI)[/TD]
[TD]$ae^{\alpha x}\cos(\beta x)+be^{\alpha x}\sin(\beta x)$[/TD]
[TD]$x^s\left(Ae^{\alpha x}\cos(\beta x)+Be^{\alpha x}\sin(\beta x) \right)$[/TD]
[/TR]
[TR]
[TD](VII)[/TD]
[TD]$p_ne^{\alpha x}\cos(\beta x)+q_me^{\alpha x}\sin(\beta x)$[/TD]
[TD]$x^se^{\alpha x}\left(P_N(x)\cos(\beta x)+Q_N(x)\sin(\beta x) \right)$
where $N=\max(n,m)$[/TD]
[/TR]
[/TABLE]

Notes:

The non-negative integer $s$ is chosen to be the smallest integer so that no term in the particular solution $y_p(x)$ is a solution to the corresponding homogeneous solution $L[y](x)=0$.

$P_n(x)$ must include all its terms even if $p_n(x)$ has some terms that are zero.

To show this, it suffices to work with type VII functions--that is, functions of the form:

(1) $$g(x)=p_n(x)e^{\alpha x}\cos(\beta x)+q_m(x)e^{\alpha x}\sin(\beta x)$$

where $p_n$ and $q_m$ are polynomials of degrees $n$ and $m$ respectively--since the other types listed in the table are just special cases of (1).

Consider the inhomogeneous equation:

(2) $$L[y](x)=g(x)$$

where $L$ is the linear operator:

(3) $$L\equiv a_nD^{n}+a_{n-1}D^{n-1}+\cdots+a_1D+a_0$$

with $a_n,\,a_{n-1},\,\cdots\,a_0$ constants, and $g(x)$ as given in equation (1). Let $N=\max(n,m)$.

Now, we need to find an annihilator for $g$. If we consider the function:

$$f(x)=e^{\alpha x}\sin(\beta x)$$

we find that:

$$f'(x)=e^{\alpha x}\left(\alpha\sin(\beta x)+\beta\cos(\beta x) \right)$$

$$f''(x)=e^{\alpha x}\left(\left(\alpha^2-\beta^2 \right)\sin(\beta x)+2\alpha\beta\cos(\beta x) \right)$$

If we observe that:

$$f''(x)-2\alpha f'(x)+\left(\alpha^2+\beta^2 \right)f(x)=0$$

then we may state that:

$$D^2-2\alpha D+\alpha^2+\beta^2=(D-\alpha)^2+\beta^2$$

annihilates $f(x)$, and so we conclude that:

$$A\equiv\left((D-\alpha)^2+\beta^2 \right)^{N+1}$$

annihilates $g$.

Now, we need to find the auxiliary equation associated with:

$$AL[y]=0$$

$$\left((D-\alpha)^2+\beta^2 \right)^{N+1}\left(a_nD^n+a_{n-1}D^{n-1}+\cdots+a_0 \right)=0$$

Suppose $0\le s$ is the multiplicity of the roots $\alpha\pm\beta i$ of the auxiliary equation associated with $L[y]=0$, and $r=2_{2s+1}\,\cdots\,r_n$ are the remaining roots, then we have:

(4) $$\left((r-\alpha)^2+\beta^2 \right)^{s+N+1}\left(r-r_{2s+1} \right)\,\cdots\,\left(r-r_n \right)=0$$

Now, as the solution to $$AL[y]=0$$ can be written in the form:

$$y(x)=y_h(x)+y_p(x)$$

and we must have:

$$y_h(x)=p_{s+1}(x)e^{\alpha x}\left(\cos(\beta x)+\sin(\beta x) \right)+\sum_{k=2s+1}^n e^{k}$$

then we may conclude that:

$$y_p(x)=x^se^{\alpha x}\left(P_N(x)\cos(\beta x)+Q_N\sin(\beta x) \right)$$

Questions and comments should be posted here:

http://mathhelpboards.com/commentary-threads-53/commentary-justifying-method-undetermined-coefficients-4840.html
 
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This topic is for questions and comments pertaining to this tutorial:

http://mathhelpboards.com/math-notes-49/justifying-method-undetermined-coefficients-4839.html
 
Good stuff! Just a couple of minor edits I'd recommend:

1. In the table, in the $y_{p}(x)$ column for Type I's, an $x^{s}$ seems to have become an $x^{2}$.

2. I would rewrite Equation (3) as follows (you haven't really used operator notation, but have included the test function in your definition of $L$, which is not usual):
$$(3) \quad L[y] \equiv a_nD^{n}+ a_{n-1}D^{n-1}+ \cdots+ a_{1}D+ a_0.$$
You do this later on, so this is more of a consistency thing, I think.
 
Thank you! For some reason, I want to enter a 2 instead of an s. I appreciate you catching this!

Your suggestion of being consistent with operator notation is an excellent one.

I have made both edits. (Sun)
 
Hehe. Actually, I'm not sure I was consistent just then! You could either write
$$L \equiv a_{n}D^{n}+a_{n-1}D^{n-1}+ \dots + a_{1}D + a_{0},$$
or
$$L[y] = \left( a_{n}D^{n}+a_{n-1}D^{n-1}+ \dots + a_{1}D + a_{0} \right)y.$$
 
I prefer the first notation. This was a group project taken from my old ODE textbook, and the original notation came from there (notice how I am "passing the buck?"). (Rofl)

I truly appreciate your suggestions, and feel the post has been improved because of them. (Rock)
 
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