Nonhomogeneous Equations and Particular Solutions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Guessing the guest that matches the forcing

Mathematics · Differential Equations · ages 20-21
Name ______________________   Date ____________

  1. y double prime minus 4 y equals 7 e to the 2 x, with homogeneous roots 2 and minus 2. Which trial fits?

    • A times e to the 2 x
    • A times x squared times e to the 2 x
    • A times x times e to the 2 x

  2. Variation of parameters applies even when the forcing has no simple guess, such as tan x. True or false?

    Circle one:   True   False

  3. Forcing is 2x plus 1 and 0 is not a homogeneous root. Which trial fits?

    • A times x squared plus B times x
    • A times e to the x
    • A times x plus B
    • A sin x plus B cos x

  4. y double prime minus 4 y equals 3 e to the 5 x, with homogeneous roots 2 and minus 2. Which trial fits?

    • A times e to the 5 x
    • A times x times e to the 5 x
    • A cosine 5 x

  5. You can choose a correct trial without knowing the homogeneous roots.

    Circle one:   True   False

  6. No menu guess fits the forcing in front of you. What is the fallback?

    • Variation of parameters
    • Guessing zero for the particular part
    • Deleting the forcing term

  7. For y double prime minus 9 y equals e to the 3 x, why does the guess A e to the 3 x fail?

    • Exponentials are banned as trials
    • The forcing is too small to change anything
    • It duplicates a homogeneous solution

  8. y double prime plus 9y equals cos 3x. Homogeneous roots are 3i and minus 3i. Which trial fits?

    • A cos 3x plus B sin 3x
    • x times (A cos 3x plus B sin 3x)
    • A cos 3x
    • A x cos 3x

  9. A student multiplies every trial by x just to be safe, even when nothing clashes. What goes wrong?

    • Extra x factors are never allowed in a trial
    • Without the plain term the trial cannot fit
    • It changes the homogeneous roots of the equation

  10. The equation has repeated homogeneous root 2 and forcing e to the 2 x. How many x factors does the trial need?

    • None
    • Two, one per repeated root
    • One, since duplication is binary
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Guessing the guest that matches the forcing W1-mt_gbkQZte_JZ-s1

  1. A times x times e to the 2 x · 2 is a homogeneous root, so the plain guess duplicates and needs one x. The root is not repeated, so one x is enough.
  2. True · It builds the particular solution from integrals of the homogeneous pieces, so any continuous forcing works.
  3. A times x plus B · Polynomial forcing of degree 1 takes a general degree 1 polynomial trial.
  4. A times e to the 5 x · Rate 5 duplicates neither root, so the plain exponential guess works.
  5. False · Only the roots reveal clashes, so skipping them builds trials to fail.
  6. Variation of parameters · Varying the parameters always works, needing only the homogeneous solutions.
  7. It duplicates a homogeneous solution · The roots are 3 and minus 3, so A e to the 3 x already satisfies the homogeneous equation, leaving nothing to balance the forcing.
  8. x times (A cos 3x plus B sin 3x) · cos 3x and sin 3x already solve the homogeneous equation, so the sine plus cosine pair must be multiplied by x.
  9. Without the plain term the trial cannot fit · With no clash, the x-multiplied trial misses the plain term. For 5e to the 3x with roots -1 and -2, A x e to the 3x gives 9A e to the 3x + 20A x e to the 3x, and no A removes the x term.
  10. Two, one per repeated root · Multiply by x repeatedly until fresh: a double root needs x squared times the exponential.
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