Method of Undetermined Coefficients

The method of undermined coefficients is an algorithm to determine the particular solution $y\_p$ to a nohomogenous differential equation.

Requirements

$G(x)$ is a sum of terms of various polynomials multiplying an exponential with sin or cos factors. That is, $G(x)$ is a sum of therms of the following forms:

• $p\_1(x)e^{rx}$
• $p\_2(x)e^{\alpha x}cos\beta x$
• $p\_3(x)e^{\alpha x}sin(\beta x)$ Examples include: $1-x,e^{2x},x{e}^x,cos(x),5e^x-sin2x$

Workflow

One considers a possible polynomial or function with unknown coefficients that could match the structure of the solution. Such could include:

• Guessing the degree of $y$ and with that $y\_p$ by considering the form of the initial differential equation and the related solution $y\_c$.

Polynomial Solution

The solving process is demonstrated on a polynomial that has been decided to represent the structure of $y\_p$

$$y\_p\_trial=A{x}^n+B{x}^{n-1}+...+C$$

This polynomial’s derivatives are found and plugged into the inital differential equation of the problem as the function $y$.

$$\text{Initial Example Equation:}{y}^{\prime \prime }+y^{\prime }+y=G(x)$$ $$y\_{p\_trial}^{\prime \prime }+y\_{p\_trial}^{\prime }+y\_p\_trial=G(x)$$

If $G(x)$ is a polynomial (this is a requirement), then terms of equal degree can be equated. An example is given with $G(x)=1-x^2$.

$$3A{x}^2+(-4A-3B)x+(2A-2B-3C)=1-x^2$$

Equating terms yields

$$3A=-1,-4A-3B=0,2A-2B-3C=1$$

Coefficients thus can be determined and plugged into the initial trial function $y\_p\_trial$ and then inserted into the final solution $y=y\_c+y\_p\_trial$ to be tested for success.

Table of Trial Function Terms

If G(x) has a term that is a constant multiple of…	And if…	Then include this expression in the trial function for yₚ
eʳˣ	r is not a root of the auxiliary equation	A eʳˣ
	r is a single root of the auxiliary equation	A x eʳˣ
	r is a double root of the auxiliary equation	A x² eʳˣ
sin kx, cos kx	ki is not a root of the auxiliary equation	B cos kx + C sin kx
p x² + qx + m	0 is not a root of the auxiliary equation	D x² + E x + F
	0 is a single root of the auxiliary equation	D x³ + E x² + F x
	0 is a double root of the auxiliary equation	D x⁴ + E x³ + F x²