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Particular Solution Form (Undetermined Coefficients)

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

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y_{p} = x^{s} e^{α x} [A_{n} (x) cos (β x) + B_{n} (x) sin (β x)]

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Core idea

Overview

The Method of Undetermined Coefficients is used to find a particular solution y_p 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 y_p based on the form of g(x), including a crucial modification factor x^s to handle cases where g(x) is a solution to the homogeneous equation.

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.

Symbols

Variables

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

g (x)

Form of g(x)

V a r iab l e

n

Degree of P_n(x)

V a r iab l e

α

Alpha (exponent)

V a r iab l e

β

Beta (frequency)

V a r iab l e

r

Homogeneous Characteristic Roots

V a r iab l e

y_{p}

Particular Solution Form

V a r iab l e

Walkthrough

Derivation

Formula: Form of Particular Solution (Undetermined Coefficients)

The form of the particular solution $y_{p}$ is chosen to mimic the non-homogeneous term g(x), with modifications if g(x) is a solution to the homogeneous equation.

The ODE is linear with constant coefficients.
The non-homogeneous term g(x) is a polynomial, exponential, sine, cosine, or a product of these.
The homogeneous solution $y_{h}$ has been found.

Initial Guess for y_p:

The initial guess for $y_{p}$ is constructed to be a linear combination of all linearly independent functions that appear in g(x) and its derivatives. This form ensures that all terms generated by differentiation can be matched with terms in g(x).

If g (x) = P_{n} (x) e^{α x} cos (β x) or P_{n} (x) e^{α x} sin (β x), the initial guess for y_{p} is e^{α x} [A_{n} (x) cos (β x) + B_{n} (x) sin (β x)]

Handling Duplication (The 's' Factor):

If a term in the initial guess for $y_{p}$ is already part of the homogeneous solution $y_{h}$ , it will result in zero when substituted into the homogeneous ODE, preventing it from satisfying the non-homogeneous equation. To resolve this, we multiply the initial guess by the smallest integer power of x, denoted as $x^{s}$ , such that no term in the modified guess is a solution to the homogeneous equation.

If any term in the initial guess for y_{p} is a solution to the homogeneous equation, then the initial guess must be multiplied by x^{s} .

Determining 's' (Multiplicity Rule):

More formally, 's' is the multiplicity of the complex number $α$ + i $β$ (derived from the exponential and trigonometric parts of g(x)) as a root of the characteristic equation of the homogeneous ODE. If $α$ + i $β$ is not a root, s=0. If it's a single root, s=1. If it's a double root, s=2 (for second-order ODEs).

s = multiplicity of (α + i β) as a root of the characteristic equation

Final Form of Particular Solution:

Combining these rules, the particular solution takes this final form, where $A_{n}$ (x) and $B_{n}$ (x) are general polynomials of degree 'n' (the degree of $P_{n}$ (x) in g(x)) with undetermined coefficients that will be found by substitution into the original ODE.

y_{p} = x^{s} e^{α x} [A_{n} (x) cos (β x) + B_{n} (x) sin (β x)]

Result

y_{p} = x^{s} e^{α x} [A_{n} (x) cos (β x) + B_{n} (x) sin (β x)]

Source: Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems, Chapter 3.5 (Method of Undetermined Coefficients)

Free formulas

Rearrangements

Solve for $s$

Particular Solution Form: Identify 's'

s = k

To identify 's' from a given particular solution form $y_{p}$ , observe the power of x that multiplies the entire exponential/trigonometric part.

Difficulty: 3/5

Solve for $α$

Particular Solution Form: Identify $α$

α = coefficient of x in the exponent of e^{α x}

To identify $α$ from the particular solution form $y_{p}$ , locate the exponent of the exponential term $e^{α x}$ .

Difficulty: 2/5

Solve for $β$

Particular Solution Form: Identify $β$

β = coefficient of x in the argument of cos (β x) or sin (β x)

To identify $β$ from the particular solution form $y_{p}$ , locate the argument of the sine or cosine terms.

Difficulty: 2/5

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Visual intuition

Graph

The graph displays an exponential growth pattern as s increases, as s acts as a multiplier in the exponent of the x term. This curve starts at a positive value and rises rapidly, illustrating how the multiplicity factor scales the polynomial solution. For a student, this means that higher multiplicity values lead to significantly larger particular solutions, where small changes in s result in dramatic shifts in the output. The most important feature is that the curve never reaches zero, meaning that any root multip

Graph type: exponential

Why it behaves this way

Intuition

Imagine constructing a 'mirror image' of the external forcing function g(x) for your solution, but if g(x) happens to match the system's natural 'rhythm,' you add an extra 'twist' ( $x^{s}$ )

y_{p}

The particular solution component of the non-homogeneous linear ordinary differential equation.

This part of the solution specifically addresses and 'responds' to the external forcing function, g(x), that makes the equation non-homogeneous.

x^{s}

A modification factor, where 's' is the smallest non-negative integer (0, 1, 2, ...) such that no term in the trial particular solution duplicates a term in the complementary

This factor acts as a 'resonance adjustment.' If the forcing function g(x) would naturally produce a solution that's already part of the system's unforced behavior,

x^{s}

ensures the trial solution is unique and captures

e^{α x}

An exponential factor in the trial particular solution, determined by the exponential component of the non-homogeneous term g(x).

This term dictates the overall growth or decay behavior of the particular solution, mirroring any exponential growth or decay present in the external forcing.

α

The real part of the complex root (or the exponent) associated with the exponential and/or sinusoidal terms in the non-homogeneous function g(x).

This value directly controls the rate at which the particular solution either grows or decays exponentially, reflecting the nature of the external influence.

A_{n} (x)

A polynomial of degree 'n' (where 'n' is the highest degree of any polynomial factor in g(x)), containing undetermined coefficients.

This polynomial component allows the particular solution to capture and match any polynomial behavior present in the external forcing function g(x).

B_{n} (x)

A polynomial of degree 'n' (where 'n' is the highest degree of any polynomial factor in g(x)), containing undetermined coefficients.

Similar to

A_{n}

(x), this polynomial captures the polynomial part of the forcing function g(x) when it's multiplied by a sine term.

cos (β x)

A cosine component in the trial particular solution, derived from any sinusoidal terms (sine or cosine) in the non-homogeneous term g(x).

This term allows the particular solution to mimic and respond to any oscillatory behavior present in the external forcing function.

sin (β x)

A sine component in the trial particular solution, derived from any sinusoidal terms (sine or cosine) in the non-homogeneous term g(x).

This term, paired with the cosine term, ensures the particular solution can fully capture the phase and amplitude of any oscillatory external forcing.

Free study cues

Insight

Canonical usage

This equation provides the structural form for a particular solution, requiring dimensional consistency across all terms with the dependent variable $y_{p}$ .

Common confusion

Students often forget that the arguments of exponential and trigonometric functions must be dimensionless, leading to incorrect units for $α$ and $β$ .

Dimension note

The exponents (s and $α$ x) and the arguments of trigonometric functions ( $β$ x) must be dimensionless for mathematical consistency.

Unit systems

$y_{p}$ Matches the dependent variable y of the ODE · The units of the particular solution y_p must be consistent with the units of the dependent variable y in the original differential equation.

$x$ Any unit of the independent variable (e.g., s, m, dimensionless) · The independent variable x can represent time, position, or any other quantity. Its units determine the units of \alpha and \beta.

$s$ Dimensionless · The exponent s is a non-negative integer determined by the multiplicity of \alpha + i\beta as a root of the characteristic equation.

$α$ Inverse of x's unit (e.g., s^-1, m^-1, dimensionless) · For the exponent \alpha x to be dimensionless, \alpha must have units inverse to x. If x is dimensionless, \alpha is also dimensionless.

$β$ Inverse of x's unit (e.g., s^-1, m^-1, dimensionless) · For the argument \beta x of the trigonometric functions to be dimensionless, \beta must have units inverse to x. If x is dimensionless, \beta is also dimensionless.

$A_{n} (x), B_{n} (x)$ Matches y_p's unit · These are polynomials whose coefficients (the undetermined coefficients) must be chosen such that the entire terms A_n(x) \cos(\beta x) and B_n(x) \sin(\beta x) have the same units as y_p.

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 $y_{p}$ .

Form of g(x)exp

Alpha (exponent)2

Beta (frequency)0

Degree of P_n(x)0

Homogeneous Characteristic Roots1,3

Solve for: $y_{p}$

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

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Modeling forced vibrations in mechanical systems, such as a spring-mass system with an external driving force.

Study smarter

Tips

Always determine the roots of the characteristic equation of the homogeneous ODE first, as they dictate the value of 's'.
If g(x) is a sum of terms, find the particular solution form for each term separately and sum them.
Remember that $A_{n}$ (x) and $B_{n}$ (x) are general polynomials of the same degree as $P_{n}$ (x) in g(x), with undetermined coefficients.
The value of 's' is the multiplicity of $α$ + i $β$ as a root of the characteristic equation (0, 1, or 2 for second-order ODEs).

Avoid these traps

Common Mistakes

Forgetting the $x^{s}$ factor when g(x) is a solution to the homogeneous equation.
Using $A_{n}$ (x) and $B_{n}$ (x) as constants when $P_{n}$ (x) is a polynomial of degree > 0.
Incorrectly identifying $α$ and $β$ from g(x).

Common questions

Frequently Asked Questions

The form of the particular solution y_p is chosen to mimic the non-homogeneous term g(x), with modifications if g(x) is a solution to the homogeneous equation.

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.

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.

Forgetting the x^s factor when g(x) is a solution to the homogeneous equation. Using A_n(x) and B_n(x) as constants when P_n(x) is a polynomial of degree > 0. Incorrectly identifying \alpha and \beta from g(x).