Method of Undetermined Coefficients/Sum of Several Terms

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:

$\map R x = \ds \sum_{k \mathop = 1}^n \map {f_k} x$

where each of the $f_k$ is either:

a real polynomial function: $\map {f_k} x = \ds \sum_{j \mathop = 0}^m a_j x^j$ for some integer $m$

a function of the form $\map {f_k} x = A e^r x$

a function of the form $\map {f_k} x = A \cos r x + B \sin r x$

the product of a combination of the three above.

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 $\map {y_p} x$.

For each $k$, use the Method of Undetermined Coefficients to find a particular solution to the nonhomogeneous linear second order ODE:

$y + p y' + q y = \map {f_k} x$

Then from Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part, all that remains to be done is to add them all up.

$\blacksquare$

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 $\text{(iv)}$