Resonance Modified Trial Factor: Clear Homogeneous Overlap

On this page: The Principle | Conditions | Failure Modes | EE Questions | Retrieval Practice | Practice Ground | Solve a Problem | Related Guides | FAQ

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The Principle

The move: when the proposed particular-solution trial duplicates homogeneous solution terms with known overlap multiplicity $s$, multiply the whole trial family by $x^s$ before solving for coefficients.

The invariant: the trial keeps the same forcing-driven base family, but the added factor $x^s$ makes the ansatz independent of the homogeneous terms already used by the zero-forcing equation.

Pattern:

$$y_p^{trial}\quad \Longrightarrow \quad x^s y_p^{trial}$$

Legal route	Illegal route
$y_h=c_1e^{2x},\ y_p^{trial}=Ae^{2x},\ s=1 \Longrightarrow y_p^{trial}=Axe^{2x}$	$y_h=c_1e^x+c_2e^{-x},\ y_p^{trial}=Ae^{2x},\ s=0 \Longrightarrow y_p^{trial}=Axe^{2x}$

The illegal route applies a resonance repair when there is no homogeneous overlap. That adds an unnecessary factor of $x$ and turns a nonresonant trial into the wrong minimal undetermined-coefficients trial.

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Conditions of Applicability

Condition: trial family overlaps homogeneous family with known overlap multiplicity s

Before applying, check: identify the trial family, compare it with the homogeneous solution family, and confirm the overlap multiplicity $s$ before multiplying by $x^s$.

If the condition is violated: multiplying by an unnecessary or wrong power of $x$ can create a trial that no longer matches the minimal undetermined-coefficients repair.

• Use $s=1$ when the trial family duplicates one homogeneous level, such as $e^{2x}$ already appearing in $y_h$.
• Use $s=2$ when both the trial and $x$ times the trial already sit in the homogeneous family, such as $e^{2x}$ and $xe^{2x}$.
• Do not use this move to choose the original forcing-based trial. That job belongs to Undetermined Coefficients Trial Family.

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

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Common Failure Modes

Failure mode: keep the resonant trial unchanged → the trial duplicates a homogeneous solution, so substitution gives no new particular-solution information.

Debug: ask whether any term in the trial family already appears in $y_h$ before solving for coefficients.

Failure mode: multiply by $x$ once when the known multiplicity is $s=2$ → the modified trial still overlaps the repeated-root homogeneous family.

Debug: after multiplying, compare the modified family against every homogeneous term again.

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Elaborative Encoding

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

Within the Principle

• What does the exponent $s$ count when the trial family overlaps the homogeneous family?
• Why does multiplying by $x^s$ change the trial family without changing the forcing pattern it is meant to match?

For the Principle

• What fast comparison tells you whether the resonance repair is legal before coefficient solving begins?
• Why is this guide about applying a known $s$, not about choosing $s$ from scratch?

Between Principles

• How does this move extend Undetermined Coefficients Trial Family when the nonresonant trial fails?

Generate an Example

• Write one resonant trial with known $s=1$ and one near miss where there is no overlap, so the trial should not be multiplied.

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Retrieval Practice

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

State the move in one sentence: _____When a particular-solution trial overlaps the homogeneous family with known multiplicity s, multiply the whole trial by x to the s before solving for coefficients.

Write the canonical pattern: _____$y_p^{trial}\mapsto x^s y_p^{trial}$

State the canonical condition: _____trial family overlaps homogeneous family with known overlap multiplicity s

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Practice Ground

Use these exercises to build move-selection fluency. (See Self-Explanation for how to learn from worked examples.)

Procedure Walkthrough

Starting from a resonant nonresonant-style trial, clear overlap with the homogeneous family before solving coefficients.

Step	Expression	Operation
0	$y_h=c_1e^{2x}+c_2e^{-x},\quad y_p^{trial}=Ae^{2x}$	Compare the proposed trial with the homogeneous family.
1	$Ae^{2x}\in \operatorname{span}\{e^{2x},e^{-x}\}$	Detect overlap with the homogeneous term $e^{2x}$.
2	$s=1$	Use the known overlap multiplicity.
3	$y_p^{trial}=x^1Ae^{2x}$	Multiply the whole trial by $x^s$.
4	$y_p^{trial}=Axe^{2x}$	Present the repaired trial in standard form.

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Drills

Micro-Chains

Reach the repaired trial. Assume the overlap multiplicity is known.

$$y_h=c_1e^{3x}+c_2e^{-x},\quad y_p^{trial}=Ae^{3x},\quad s=1$$

Reveal

The trial overlaps the homogeneous term $e^{3x}$ once, so multiply by $x$:

$$y_p^{trial}=Axe^{3x}.$$

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Reach the repaired trial. Assume the overlap multiplicity is known.

$$y_h=c_1e^{2x}+c_2xe^{2x},\quad y_p^{trial}=Ae^{2x},\quad s=2$$

Reveal

The homogeneous family already contains $e^{2x}$ and $xe^{2x}$, so multiply by $x^2$:

$$y_p^{trial}=Ax^2e^{2x}.$$

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Reject or repair the route choice.

$$y_h=c_1\cos 4x+c_2\sin 4x,\quad y_p^{trial}=A\cos 4x+B\sin 4x,\quad s=1$$

Reveal

Repair it. The trial duplicates the homogeneous sine-cosine family, so multiply the whole family by $x$:

$$y_p^{trial}=x(A\cos 4x+B\sin 4x).$$

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Reject or complete the route choice.

$$y_h=c_1e^x+c_2e^{-x},\quad y_p^{trial}=Ae^{2x},\quad s=0$$

Reveal

Reject the resonance repair. There is no overlap with $e^{2x}$, so $s=0$ and the trial stays

$$y_p^{trial}=Ae^{2x}.$$

Multiplying by $x$ would be an unnecessary resonance modification.

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Reach the repaired trial. Assume the known overlap multiplicity is $s=1$.

$$y_h=c_1+c_2e^{-2x},\quad y_p^{trial}=A$$

Reveal

The constant trial overlaps the homogeneous constant term. Multiply by $x$:

$$y_p^{trial}=Ax.$$

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Forward Steps

Apply the resonance modified trial factor once. Assume known overlap multiplicity $s=1$.

$$y_p^{trial}=A\cos 2x+B\sin 2x$$

Reveal

Multiply the whole trial family by $x$:

$$y_p^{trial}=x(A\cos 2x+B\sin 2x).$$

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Apply the resonance modified trial factor once. Assume known overlap multiplicity $s=2$.

$$y_p^{trial}=Ae^{-x}$$

Reveal

Multiply the trial by $x^2$:

$$y_p^{trial}=Ax^2e^{-x}.$$

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A student proposes $y_p^{trial}=Axe^{5x}$. The known overlap multiplicity is $s=2$ for the original trial $Ae^{5x}$. What is missing?

Reveal

One more factor of $x$ is missing. With $s=2$, the repaired trial is

$$y_p^{trial}=Ax^2e^{5x}.$$

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Which original trials are eligible for a resonance repair? Assume the listed $s$ values are known.

A. $y_h=c_1e^x+c_2e^{-x}$, $y_p^{trial}=Ae^x$, $s=1$
B. $y_h=c_1e^x+c_2e^{-x}$, $y_p^{trial}=Ae^{3x}$, $s=0$
C. $y_h=c_1\cos x+c_2\sin x$, $y_p^{trial}=A\cos x+B\sin x$, $s=1$

Reveal

A and C are eligible because the trial families overlap the homogeneous family with positive known multiplicity.

B is a near miss: $Ae^{3x}$ does not overlap the listed homogeneous family, so no resonance repair is applied.

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Canonicalization

Rewrite the repaired trial in standard form. Assume $s=1$.

$$x^s(Ae^{2x}+Be^{2x}x)$$

Reveal

Substitute $s=1$ and distribute the outside factor:

$$x(Ae^{2x}+Bxe^{2x})=Axe^{2x}+Bx^2e^{2x}.$$

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Canonicalize the repaired trial. Assume $s=2$.

$$x^s(A\cos 3x+B\sin 3x)$$

Reveal

With $s=2$:

$$y_p^{trial}=x^2(A\cos 3x+B\sin 3x).$$

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Solve a Problem

Apply what you’ve learned with Problem Solving.

Problem: Starting from the nonresonant-style trial $y_p^{trial}=Ae^x+Bxe^x$, repair it when the homogeneous family contains $e^x$ and $xe^x$ and the known overlap multiplicity is $s=2$.

Full solution

Step	Expression	Move
0	$y_h=c_1e^x+c_2xe^x,\quad y_p^{trial}=Ae^x+Bxe^x$	Compare the trial family with the homogeneous family.
1	$Ae^x+Bxe^x\in \operatorname{span}\{e^x,xe^x\}$	Detect that both trial terms overlap homogeneous terms.
2	$s=2$	Use the stated overlap multiplicity.
3	$y_p^{trial}=x^2(Ae^x+Bxe^x)$	Multiply the whole trial by $x^s$.
4	$y_p^{trial}=Ax^2e^x+Bx^3e^x$	Put the repaired trial in standard form.

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Related Guides

• Differential Equations Subdomain - See where resonance appears in the higher-order linear sequence.
• Undetermined Coefficients Trial Family - Choose the original forcing-based trial before checking resonance.
• Linear Nonhomogeneous Solution Structure - Review why a particular solution is added to the homogeneous family.
• Repeated Root Solution Family - Connect repeated homogeneous roots to overlap multiplicity.
• Principle Structures - Treat the name, equation, and condition as separate recall targets.

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FAQ

What is Resonance Modified Trial Factor?

Resonance Modified Trial Factor is the undetermined-coefficients repair $y_p^{trial}\mapsto x^s y_p^{trial}$. It is used when the original trial overlaps the homogeneous solution family and the overlap multiplicity $s$ is already known.

Why multiply by x to the s?

The extra factor moves the trial out of the homogeneous family. If the original trial already solves the homogeneous equation, it cannot also supply an independent particular solution for the forcing term.

When is this resonance repair valid?

It is valid when the trial family overlaps the homogeneous family with known overlap multiplicity $s$. The guide assumes that $s$ has already been found; the move here is applying the factor $x^s$ to the whole trial.

What happens if I forget the factor?

The trial may stay inside the homogeneous solution family. Then substituting it into the nonhomogeneous equation can give zero contribution from the operator instead of the forcing term you need to match.

Is this the same as choosing the undetermined-coefficients trial?

No. First choose the trial family from the forcing pattern. Then check whether that trial is resonant with the homogeneous solution family. Resonance Modified Trial Factor is the repair after that overlap is known.

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How This Fits in Unisium

Within the differential equations subdomain, Unisium treats resonance as a legality check between the forcing-based trial and the homogeneous solution family. The Unisium Study System pairs that check with retrieval practice, self-explanation, and compact problem-solving chains so the repair factor becomes fluent without hiding the condition.

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

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