Undetermined Coefficients Trial Family: Match the Forcing

On this page: The Principle | Conditions | Misconceptions | EE Questions | Retrieval Practice | Worked Example | Solve a Problem | Related Principles | FAQ

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The Principle

Statement

Undetermined coefficients starts by matching the shape of the forcing term. If the forcing term is a polynomial times an exponential times a cosine, the trial particular solution keeps the same exponential and trigonometric frequency, then supplies unknown polynomial coefficients for both cosine and sine.

The matching family is:

$g(x)=P_n(x)e^{\alpha x}\cos(\beta x) \Rightarrow y_p^{trial}=e^{\alpha x}(Q_n(x)\cos(\beta x)+R_n(x)\sin(\beta x))$

This principle chooses the trial family. Solving for the unknown coefficients comes later by substituting the trial into the differential equation.

Mathematical Form

$g(x)=P_n(x)e^{\alpha x}\cos(\beta x) \Rightarrow y_p^{trial}=e^{\alpha x}(Q_n(x)\cos(\beta x)+R_n(x)\sin(\beta x))$

Where:

• $g(x)$ = the forcing term on the right side of the differential equation
• $P_n(x)$ = a known polynomial of degree $n$
• $\alpha$ = the exponential rate in the forcing
• $\beta$ = the trigonometric frequency in the forcing
• $Q_n(x)$, $R_n(x)$ = unknown polynomials of degree $n$
• $y_p^{trial}$ = the trial particular solution before coefficients are determined

Why both cosine and sine appear

When $\beta\neq 0$, differentiating cosine terms produces sine terms and differentiating sine terms produces cosine terms. Including both $Q_n(x)\cos(\beta x)$ and $R_n(x)\sin(\beta x)$ gives the constant-coefficient operator enough room to match the forcing after derivatives are taken.

This guide sits after Linear Nonhomogeneous Solution Structure. That earlier principle says a full solution has a homogeneous part plus one particular solution; this principle helps choose the particular-solution trial when the forcing has a standard shape.

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Conditions of Applicability

Condition: constant-coefficient linear equation; standard forcing family; nonresonant case

Practical modeling notes

• “Standard forcing family” means polynomial, exponential, sine, cosine, or products of those forms.
• “Nonresonant” means the proposed trial family does not duplicate a term already present in the complementary homogeneous solution.
• If the forcing is a sum of standard terms, build one trial family for each forcing family and add the trials.
• The degree of $Q_n(x)$ and $R_n(x)$ should match the degree of the polynomial factor in the forcing.
• For purely polynomial or exponential forcing, treat this as the non-trigonometric special case: the sine-cosine pair collapses, so the trial uses the matching polynomial-exponential form rather than forcing an unnecessary sine slot.

When It Doesn’t Apply

This trial selection is not enough in these cases:

• Resonance: if the trial overlaps the homogeneous family, multiply the trial by the required power of $x$ before solving for coefficients.
• Nonstandard forcing: functions such as $\ln x$, $\tan x$, or arbitrary variable-coefficient forcing usually need another method.
• Variable coefficients: undetermined coefficients depends on constant-coefficient linear operators, not general variable-coefficient equations.

Want the complete framework behind this guide? Read Masterful Learning.

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Common Misconceptions

Misconception 1: “The trial is already the particular solution”

The truth: the trial is only a family of candidates. The unknown coefficients still have to be solved by substitution.

Why this matters: writing the right family is the representation step; it does not by itself satisfy the differential equation.

Misconception 2: “A cosine forcing only needs a cosine trial”

The truth: constant-coefficient derivatives can mix sine and cosine terms, so the standard trial includes both.

Why this matters: omitting the sine part can leave no coefficient available to cancel derivative-created sine terms.

Misconception 3: “Resonance is a small detail”

The truth: resonance changes the legal trial family. If the trial overlaps the homogeneous solution, the nonresonant trial must be modified before coefficients are determined.

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Elaborative Encoding

Use these questions to build deep understanding. (See Elaborative Encoding for the full method.)

Within the Principle

• What roles do $P_n(x)$, $Q_n(x)$, and $R_n(x)$ play in the trial-family equation?
• Why does the trial preserve the same $\alpha$ and $\beta$ from the forcing term?

For the Principle

• What should you check in the homogeneous solution before accepting the nonresonant trial?
• How does the degree of the polynomial forcing decide the degree of the unknown polynomials in the trial?

Between Principles

• How does this principle supply the particular-solution piece required by Linear Nonhomogeneous Solution Structure?

Generate an Example

• Write one forcing term that fits the standard family and one forcing term that would not be a good undetermined-coefficients target.

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Retrieval Practice

Answer from memory, then click to reveal and check. (See Retrieval Practice for the full method.)

State the principle in words: _____A standard forcing family suggests a matching trial particular-solution family before the unknown coefficients are solved.

Write the canonical equation: _____$g(x)=P_n(x)e^{\alpha x}\cos(\beta x) \Rightarrow y_p^{trial}=e^{\alpha x}(Q_n(x)\cos(\beta x)+R_n(x)\sin(\beta x))$

State the canonical condition: _____constant-coefficient linear equation; standard forcing family; nonresonant case

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Worked Example

Use this worked example to practice Self-Explanation.

Problem

For the nonhomogeneous equation $y^{\prime\prime}+y=3e^x\cos 2x$, choose the undetermined-coefficients trial family for a particular solution. Assume the nonresonant condition has been checked.

Step 1: Verbal Decoding

Target: $y_p^{trial}$
Given: $g(x)$, $P_0(x)$, $\alpha$, $\beta$
Constraints: constant-coefficient linear equation; standard exponential-cosine forcing; nonresonant case; coefficients not solved yet

Step 2: Visual Decoding

Draw a two-slot forcing pattern: exponential envelope $e^x$ outside, trigonometric frequency $2$ inside. Under it, sketch two matching slots for cosine and sine with unknown coefficients. (The trial keeps the envelope and frequency but adds room for both trig parts.)

Step 3: Mathematical Modeling

1. $g(x)=3e^x\cos 2x$
2. $y_p^{trial}=e^x(Q_0(x)\cos 2x+R_0(x)\sin 2x)$

Step 4: Mathematical Procedures

1. $Q_0(x)\cos 2x+R_0(x)\sin 2x=A\cos 2x+B\sin 2x$
2. $\underline{y_p^{trial}=e^x(A\cos 2x+B\sin 2x)}$

Step 5: Reflection

• Verification: the trial has the same exponential rate and trig frequency as the forcing term.
• Connection to concept: the sine term is included even though the forcing displays only cosine.
• Domain check: the prompt states the nonresonant case, so no extra factor of $x$ is needed here.

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Before moving on: self-explain the model

Try explaining Step 3 out loud (or in writing): why the forcing pattern, not the differential operator alone, determines the trial family, and why checking nonresonance matters before coefficient solving begins.

Mathematical model with explanation

Principle: Undetermined Coefficients Trial Family - $g(x)=P_n(x)e^{\alpha x}\cos(\beta x) \Rightarrow y_p^{trial}=e^{\alpha x}(Q_n(x)\cos(\beta x)+R_n(x)\sin(\beta x))$.

Conditions: constant-coefficient linear equation; standard forcing family; nonresonant case.

Relevance: the right side $3e^x\cos 2x$ is a standard exponential-cosine forcing term, so the trial family can be read from its shape.

Description: The forcing has polynomial degree $0$, exponential rate $1$, and trigonometric frequency $2$. The trial keeps those features and inserts two unknown constants, one for cosine and one for sine.

Goal: choose the correct trial family before substituting into the differential equation to solve for $A$ and $B$.

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Solve a Problem

Apply what you’ve learned with Problem Solving.

Problem

For the nonhomogeneous equation $y^{\prime\prime}+4y=(x-1)e^{-x}\cos 3x$, choose the undetermined-coefficients trial family for a particular solution. Assume the nonresonant condition has been checked.

Hint (if needed): the polynomial factor has degree $1$, so both unknown polynomial factors in the trial should have degree $1$.

Show Solution

Step 1: Verbal Decoding

Target: $y_p^{trial}$
Given: $g(x)$, $P_1(x)$, $\alpha$, $\beta$
Constraints: constant-coefficient linear equation; standard polynomial-exponential-cosine forcing; nonresonant case; coefficients not solved yet

Step 2: Visual Decoding

Draw the forcing as three stacked features: degree-one polynomial, exponential envelope $e^{-x}$, and cosine frequency $3$. Then draw the trial with the same envelope and frequency, but with degree-one unknown polynomials on both cosine and sine. (The degree-one forcing creates degree-one unknown coefficient polynomials.)

Step 3: Mathematical Modeling

1. $g(x)=(x-1)e^{-x}\cos 3x$
2. $y_p^{trial}=e^{-x}(Q_1(x)\cos 3x+R_1(x)\sin 3x)$

Step 4: Mathematical Procedures

1. $Q_1(x)\cos 3x+R_1(x)\sin 3x=(Ax+B)\cos 3x+(Cx+D)\sin 3x$
2. $\underline{y_p^{trial}=e^{-x}((Ax+B)\cos 3x+(Cx+D)\sin 3x)}$

Step 5: Reflection

• Verification: the trial keeps the same exponential rate, frequency, and polynomial degree as the forcing.
• Connection to concept: both sine and cosine receive degree-one polynomial coefficients.
• Domain check: this is the stated nonresonant case, so the trial is not multiplied by an extra power of $x$.

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Related Principles

Principle	Relationship to Undetermined Coefficients Trial Family
Linear Nonhomogeneous Solution Structure	Explains why the trial particular solution is added to the homogeneous family.
Resonance Modified Trial Factor	Covers the next case, where overlap with the homogeneous family forces an extra factor of $x$.
Complex Roots Solution Family	Helps interpret exponential-sine-cosine families that also appear in homogeneous solutions.

See Differential Equations Subdomain for the full higher-order linear sequence, and Principle Structures for organizing trial-family conditions next to nearby solution structures.

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FAQ

What is Undetermined Coefficients Trial Family?

Undetermined Coefficients Trial Family is the rule that a standard forcing term suggests a matching trial family for a particular solution. For $P_n(x)e^{\alpha x}\cos(\beta x)$, the nonresonant trial is $e^{\alpha x}(Q_n(x)\cos(\beta x)+R_n(x)\sin(\beta x))$.

When does Undetermined Coefficients Trial Family apply?

It applies to constant-coefficient linear equations with a standard forcing family in the nonresonant case. The forcing should be built from polynomials, exponentials, sines, cosines, or their standard products.

Why include sine if the forcing only has cosine?

Derivatives of sine and cosine feed into each other. Including both terms gives the trial enough freedom to match the forcing after the differential operator is applied.

What does nonresonant mean here?

Nonresonant means the proposed trial family does not duplicate a term in the homogeneous solution family. If it does duplicate one, the trial must be multiplied by the appropriate power of $x$ before coefficients are solved.

Does this principle solve for the coefficients?

No. It chooses the candidate family. After the trial is chosen, you substitute it into the differential equation and solve for the unknown coefficients.

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Related Guides

• Differential Equations Subdomain - Return to the full DE map and see where undetermined coefficients sits.
• Linear Nonhomogeneous Solution Structure - Review why one particular solution combines with the homogeneous family.
• Retrieval Practice - Make the trial family and condition quick to recall.
• Problem Solving - Practice choosing the model before doing coefficient algebra.

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How This Fits in Unisium

Within the differential equations subdomain, Unisium treats Undetermined Coefficients Trial Family as the decision point where the forcing term becomes a candidate particular-solution shape. The Unisium Study System pairs this guide with elaborative encoding, retrieval practice, and structured problem solving so you learn to choose the family before calculating coefficients.

Ready to practice differential equations with structure? Check access and join the Unisium waitlist or explore the full framework in Masterful Learning.

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