Method of Undetermined Coefficients/Polynomial
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
- 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 2$. The second order equation: $\S 2.3$ Particular solution: polynomial $\map f x$
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.18$: The Method of Undetermined Coefficients