Solving Initial Value Problems with Laplace Transforms · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Trade calculus for algebra

Mathematics · Differential Equations · ages 21-22
Name ______________________   Date ____________

  1. The transform of e to the two t is one over (s minus two). Evaluated at s equal three, what value results?

    Answer: ______________

  2. y(0) is 4. What is the Laplace transform of y prime?

    • s times Y
    • Y divided by s
    • s times Y minus 4
    • s squared times Y

  3. Transform y prime minus 2y equals zero, starting at y of zero equals 3. What equation holds for Y?

    • s Y minus 3 minus 2Y equals zero
    • s Y minus 2Y equals zero
    • s squared Y minus 3 minus 2Y equals zero

  4. Transforming y prime equals zero, starting at y of zero equals 5, gives Y equals 5s.

    Circle one:   True   False

  5. Splitting four divided by (s times (s plus two)) gives A over s plus B over (s plus two). Covering s, A is four divided by two. What is A?

    Answer: ______________

  6. How does s over s squared plus nine invert?

    • Cosine of three t
    • Sine of three t
    • Cosine of nine t

  7. Y(s) is three divided by (s minus two) plus one divided by (s plus one). What is y(t)?

    • Three e to the two t plus e to minus t
    • Three e to minus two t plus e to the t
    • Three t squared plus one
    • Three cosine plus sine

  8. A student transforms u of (t minus two) times sin(t minus two) to e to minus two s divided by (s squared plus one). True or false?

    Circle one:   True   False

  9. Forcing of sine of t minus five meets u of t minus five. What factor appears?

    • e to plus five s
    • No factor at all
    • e to minus five s

  10. Y(s) is two divided by (s squared plus four) plus one divided by (s minus one). What is y(t)?

    • Cosine of two t plus e to the t
    • Sine of two t plus e to minus t
    • Two t plus e to the t
    • Sine of two t plus e to the t
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Answer key

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

Trade calculus for algebra W1-mt_0BhjPVNUwl-s1

  1. 1 · One over (three minus two) is one.
  2. s times Y minus 4 · Differentiation gives s times Y minus the initial value 4.
  3. s Y minus 3 minus 2Y equals zero · y prime becomes s Y minus the starting value 3, and minus 2y becomes minus 2Y, so s Y minus 3 minus 2Y equals zero.
  4. False · s Y minus 5 equals zero, so Y equals 5 over s, not 5s.
  5. 2 · Four divided by two is two.
  6. Cosine of three t · s over s squared plus k squared is cosine of k t; here k squared is nine, so k is three.
  7. Three e to the two t plus e to minus t · Each chunk inverts by sight to its exponential.
  8. True · True. The shift matches, so the second shift theorem applies directly.
  9. e to minus five s · A matched delay of five shifts the transform by e to minus five s.
  10. Sine of two t plus e to the t · Two over (s squared plus four) is sine of two t, and one over (s minus one) is e to the t.
Worksheet · LightMySky