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

Guessing the guest that matches the forcing

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

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

    Circle one:   True   False

  2. y double prime plus 3 y prime plus 2y equals 5 times e to the 3x. Homogeneous roots are minus 1 and minus 2. Which trial fits?

    • A times e to the 3x
    • A times x times e to the 3x
    • A cos 3x
    • A times x squared

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

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

  4. Same left side, but the forcing is 4 e to the minus x. Which trial fits?

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

  5. The forcing is a sine wave. What shape should the trial take?

    • A constant only
    • A sine plus cosine pair
    • A single exponential

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

    Circle one:   True   False

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

    • Variation of parameters, letting the constants vary
    • Guessing zero for the particular part
    • Deleting the forcing term

  8. For y double prime minus y equals e to the x, why does the obvious guess fail?

    • Exponentials are banned as trials
    • The forcing is too small to matter
    • It duplicates a homogeneous solution, so substitution gives zero

  9. A student multiplies every trial by x just to be safe. Why is that wasteful and risky?

    • Extra x factors are always illegal
    • It adds needless algebra and can create fresh clashes
    • It changes the homogeneous roots

  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. True · It builds the particular solution from integrals of the homogeneous pieces, so any continuous forcing works.
  2. A times e to the 3x · The forcing is exponential with rate 3, which duplicates no root, so a plain multiple works.
  3. A times x plus B · Polynomial forcing of degree 1 takes a general degree 1 polynomial trial.
  4. A times x times e to the minus x · Minus 1 is a homogeneous root, so the plain guess duplicates and needs one x.
  5. A sine plus cosine pair · Sines invite both sine and cosine with unknown coefficients to solve for.
  6. False · Only the roots reveal clashes, so skipping them builds trials to fail.
  7. Variation of parameters, letting the constants vary · Varying the parameters always works, needing only the homogeneous solutions.
  8. It duplicates a homogeneous solution, so substitution gives zero · A e to the x already satisfies the homogeneous equation, leaving nothing to balance the forcing.
  9. It adds needless algebra and can create fresh clashes · Adjust only on real duplication; blind x factors bloat the work and may clash again.
  10. Two, one per repeated root · Multiply by x repeatedly until fresh: a double root needs x squared times the exponential.
Worksheet · LightMySky