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Equations about rates, solved for the function itself.

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Builds on: Integration techniques (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Initial value problem par 30 s not tried yet

  worked example

  y′ = −6x² + 4x − 4 and y(1) = 0. What is y(2)?

  Answer: -12

  1. Integrate: y = −2x³ + 2x² − 4x + C.
  2. y(1) = −4 + C = 0, so C = 4.
  3. Then y(2) = −16 + 4 = −12.

• y′ = ky par 30 s not tried yet

  worked example

  y′ = y and y(0) = 6. What is y(ln 3)?

  Answer: 18

  1. The solution is y = 6eᵗ.
  2. At t = ln 3: e^(ln 3) = 3.
  3. So y(ln 3) = 6 × 3 = 18.

• Characteristic roots par 30 s not tried yet

  worked example

  y″ − 9y = 0 has solutions of the form eʳᵗ. What are the two values of r? Enter both, smaller first.

  Answer: (-3, 3)

  1. Substituting y = eʳᵗ gives r² − 9 = 0.
  2. This factors as (r + 3)(r − 3) = 0.
  3. So r = −3 or r = 3. Smaller first: −3, 3.

• Laplace transforms par 30 s not tried yet

  worked example

  The Laplace transform of f(t) = e⁵ᵗ is F(s). What is F(6)? Give an exact number or fraction.

  Answer: 1

  1. L{eᵃᵗ} = 1/(s − a) for s > a. Here a = 5, so F(s) = 1/(s − 5).
  2. F(6) = 1/(6 − 5) = 1.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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