Method of Undetermined Coefficients/Product of Polynomial and Function of Sine and Cosine
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 of the form:
- $\map R x = \paren {\alpha \cos b x + \beta \sin b x} \paren {\map f x}$
where $\map f x$ is a real polynomial function.
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:
- $(2): \quad y + p y' + q y = 0$
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 $\map {y_p} x$.
Substitute a trial solution of similar form, either:
- $\paren {\alpha \cos b x + \beta \sin b x} \paren {\map g x}$
or replace the right hand side of $(1)$ by:
- $\paren {\alpha - i \beta} e^{i \paren {a + i b} x} \paren {\map g x}$
find a solution, and take the real part.
In the above, $\map g x$ is a real polynomial function with undetermined coefficients of as high a degree as $f$.
Then:
- differentiate twice with respect to $x$
- establish a set of simultaneous equations by equating powers
- solve these equations for the coefficients.
If $\paren {\alpha \cos b x + \beta \sin b x} \paren {\map g x}$ appears in the general solution to $(2)$, then add a further degree to $g$.
The last step may need to be repeated if that last polynomial also appears as a general solution to $(2)$.
$\blacksquare$
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: $\S 2.6$ Particular solution: some further cases $\text{(iii)}$