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Math101learn.math101.caUndetermined Coefficients
A precise trial-function method for particular solutions of constant-coefficient linear equations.
Precise definition
Undetermined coefficients finds a particular solution of a linear constant-coefficient ODE when the forcing is a finite combination of polynomials, exponentials, sines, cosines, and their products. Choose a trial family closed under differentiation, substitute, and solve algebraically for its coefficients.
Notation and mathematical language
For forcing $e^{\alpha t}P_m(t)\cos\beta t$ or sine, include both sine and cosine with general degree-$m$ polynomials. If the trial overlaps a homogeneous mode whose characteristic root $\alpha+i\beta$ has multiplicity $s$, multiply the entire trial by $t^s$.
Conceptual picture
The operator maps the chosen finite-dimensional trial space into itself. Coefficient matching in that space replaces integration. Resonance means the operator annihilates part of the naive trial, so multiplication by $t$ moves to an independent space.
Conditions and key results
The method requires constant coefficients and an allowed forcing family. It is not a general method for $\ln t$, $\tan t$, or arbitrary data. A particular solution is not unique, but two particular solutions differ by a homogeneous solution.
A reliable strategy
- Solve the homogeneous characteristic equation and record roots with multiplicity.
- Identify the forcing family and write the full trial, including all polynomial terms and sine/cosine partners.
- Check resonance and multiply by the required power of $t$ before differentiating.
- Substitute, match coefficients, solve them, then add $y_h$ and apply any data.
Fully worked example
Interpretation and application
The method computes forced responses in circuits and vibrations. Resonant algebra reflects a physical or dynamical mode match, but real amplitude conclusions must incorporate damping, parameter uncertainty, and the forcing's duration.
Common mistakes
Verification and reasonableness
- Substitute the particular trial into the full operator and recover exactly the forcing.
- Check the trial shares no term with the homogeneous solution after adjustment.
- Differentiate carefully and match every independent polynomial, sine, and cosine coefficient.
Practice
- What trial fits forcing $t^2$ with no resonance?
- What trial fits $e^{2t}\cos3t$?
- What adjustment handles a simple resonant root?
Answers and brief solutions
- $At^2+Bt+C$.
- $e^{2t}(A\cos3t+B\sin3t)$.
- Multiply the entire naive trial by $t$.
Further deduction
For $y''+y=\cos t$, the roots are $\pm i$, so both cosine and sine lie in the homogeneous space. The resonant trial is $t(A\cos t+B\sin t)$. A shorter equivalent choice may emerge after substitution, but starting with the complete adjusted family prevents accidentally excluding the needed component.
A quick operator check can reduce arithmetic. For $L(D)e^{\alpha t}=L(\alpha)e^{\alpha t}$ when $L(\alpha)\ne0$, a forcing $Ce^{\alpha t}$ has particular solution $[C/L(\alpha)]e^{\alpha t}$. If $L(\alpha)=0$, resonance is present and the shortcut correctly signals failure. The full trial method remains safer when polynomial or trigonometric factors appear.
The trial family must also be closed under differentiation. A polynomial forcing needs a full polynomial of the same degree because differentiating mixes its coefficients; using only the highest-power term cannot generally match all lower powers created by the differential operator.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
If forcing is $e^{2t}$ and $r=2$ is a simple characteristic root, what power of $t$ multiplies the trial?
- The naive trial $Ae^{2t}$ is homogeneous.
- For multiplicity one, use $At e^{2t}$, so the power is 1.
End of lesson
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