Method of Undetermined Coefficients

Definition

The Method of Undetermined Coefficients is a heuristic for finding $y_p$ by guessing a form similar to the forcing function $G(x)$ and solving for unknown constants.

• How to read: “The particular solution y subscript p, and the nonhomogeneous term G of x.”
• Meaning: $y_p$ is the particular solution; $G(x)$ is the nonhomogeneous forcing term on the right side of the ODE.

Why It Matters

Failing to master this method forces engineers to rely on more complex, computationally expensive techniques for simple linear systems. In fields like AC circuit design or mechanical vibration analysis, this inefficiency increases the risk of calculation errors and delays in modeling steady-state responses.

Core Concepts

• Matching Forms:

  • If $G(x)$ is a polynomial of degree $n$, guess $y_p = A_nx^n + \dots + A_0$.

    • How to read: “The particular solution y subscript p is a polynomial of degree n, from A subscript n times x to the n, down to the constant term A subscript zero.”
    • Meaning: Match polynomial degree in the guess to the forcing polynomial.

  • If $G(x)$ is $e^{kx}$, guess $y_p = Ae^{kx}$.

    • How to read: “The particular solution y subscript p is equal to the constant A times e raised to the power of k times x.”
    • Meaning: Exponential forcing gets an exponential particular solution with undetermined amplitude $A$.

  • If $G(x)$ is $\sin kx$ or $\cos kx$, guess $y_p = A\cos kx + B\sin kx$.

    • How to read: “The particular solution y subscript p is equal to the constant A times the cosine of k times x, plus the constant B times the sine of k times x.”
    • Meaning: Trig forcing requires both sine and cosine in the guess to handle phase.

• Modification Rule: If the guess (or any part of it) is already a solution to the homogeneous equation ($y_c$), multiply the guess by $x$ or $x^2$ until it is linearly independent.

• Algebraic Determination: Substitute the guess into the ODE and equate coefficients of like terms to find the “undetermined” constants.

Connected Concepts

• The Closed Interval Method
• Euler’s Method
• Fourier Coefficients