Method of Undetermined Coefficients/Polynomial

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Proof Technique

Consider the nonhomogeneous linear second order ODE with constant coefficients:

$(1): \quad y'' + p y' + q y = \map R x$

Let $\map R x$ be a polynomial in $x$:

$\ds \map R x = \sum_{j \mathop = 0}^n a_j x^j$


The Method of Undetermined Coefficients can be used to solve $(1)$ in the following manner.


Method and Proof

Let $\map {y_g} x$ be the general solution to:

$y'' + p y' + q y = 0$

From Solution of Constant Coefficient Homogeneous LSOODE, $\map {y_g} x$ can be found systematically.

Let $\map {y_p} x$ be a particular solution to $(1)$.

Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:

$\map {y_g} x + \map {y_p} x$

is the general solution to $(1)$.


It remains to find $\map {y_p} x$.


Let $\ds \map R x = \sum_{j \mathop = 0}^n a_j x^j$.


Assume that there is a particular solution to $(1)$ of the form:

$\ds y_p = \sum_{j \mathop = 0}^n A_j x^j$

We have:

\(\ds \frac {\d} {\d x} y_p\) \(=\) \(\ds \sum_{j \mathop = 1}^n j A_j x^{j - 1}\) Power Rule for Derivatives
\(\ds \frac {\d^2} {\d x^2} y_p\) \(=\) \(\ds \sum_{j \mathop = 2}^n j \paren {j - 1} A_j x^{j - 2}\) Power Rule for Derivatives


Inserting into $(1)$:

\(\ds \sum_{j \mathop = 2}^n j \paren {j - 1} A_j x^{j - 2} + p \sum_{j \mathop = 1}^n j A_j x^{j - 1} + q \sum_{j \mathop = 0}^n A_j x^j\) \(=\) \(\ds \sum_{j \mathop = 0}^n a_j x^j\)

The coefficients $A_0$ to $A_n$ can hence be solved in terms of $a_0$ to $a_n$ using the techniques of simultaneous equations.

The general form is tedious and unenlightening to evaluate.

$\blacksquare$


Sources