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Particular Solution Form (Undetermined Coefficients) Calculator

Determines the correct trial form for the particular solution of a non-homogeneous linear ODE.

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Particular Solution Form

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Overview

The Method of Undetermined Coefficients is used to find a particular solution for non-homogeneous linear ordinary differential equations with constant coefficients, specifically when the non-homogeneous term g(x) is of a certain form (polynomials, exponentials, sines, cosines, or products thereof). This formula provides the general structure of the trial solution based on the form of g(x), including a crucial modification factor to handle cases where g(x) is a solution to the homogeneous equation.

Symbols

Variables

g(x) = Form of g(x), n = Degree of P_n(x), \alpha = Alpha (exponent), \beta = Beta (frequency), r = Homogeneous Characteristic Roots

Form of g(x)
Degree of P_n(x)
Alpha (exponent)
Beta (frequency)
Homogeneous Characteristic Roots
Particular Solution Form

Apply it well

When To Use

When to use: Apply this formula when solving non-homogeneous linear ODEs with constant coefficients where the g(x) term is a polynomial, exponential, sine, cosine, or a product of these. It's the first step in the Method of Undetermined Coefficients, before differentiating and substituting to find the actual coefficients.

Why it matters: Mastering this formula is essential for solving a wide class of non-homogeneous differential equations, which model numerous phenomena in physics, engineering, and economics (e.g., forced oscillations, RLC circuits). Correctly identifying the form of y_p is critical for the success of the entire method, as an incorrect form will lead to a failed solution.

Avoid these traps

Common Mistakes

  • Forgetting the factor when g(x) is a solution to the homogeneous equation.
  • Using (x) and (x) as constants when (x) is a polynomial of degree > 0.
  • Incorrectly identifying and from g(x).

One free problem

Practice Problem

For a non-homogeneous ODE, the forcing term is g(x) = 3e^(2x). The characteristic roots of the corresponding homogeneous equation are r1=1 and r2=3. Determine the correct form of the particular solution .

Form of g(x)exp
Alpha (exponent)2
Beta (frequency)0
Degree of P_n(x)0
Homogeneous Characteristic Roots1,3

Solve for:

Hint: Identify and from g(x), then check if + i is a root of the characteristic equation to find 's'.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Boyce, DiPrima, and Meade: Elementary Differential Equations and Boundary Value Problems
  2. Zill and Wright: Differential Equations with Boundary-Value Problems
  3. Wikipedia: Method of undetermined coefficients
  4. Boyce, W. E., DiPrima, R. C., & Meade, D. B. (2017). Elementary Differential Equations and Boundary Value Problems (11th ed.).
  5. Zill, D. G. (2017). A First Course in Differential Equations with Modeling Applications (11th ed.). Cengage Learning.
  6. Wikipedia: Dimensional analysis
  7. Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima
  8. Differential Equations with Boundary-Value Problems by Zill