First-Order Linear Equations and the Integrating Factor · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The factor that builds a product

Mathematics · Differential Equations · ages 19-20
Name ______________________   Date ____________

  1. Multiplying by the integrating factor turns the left side into the derivative of the factor times y. True or false?

    Circle one:   True   False

  2. Put 2 times y prime plus 4y equals 6 into standard form y prime plus P times y equals Q. What is it?

    • y prime plus 2y equals 3
    • y prime plus 4y equals 6
    • 2y prime plus 2y equals 3
    • y prime plus 2y equals 6

  3. For y prime plus (2 over x) times y equals 1, the integrating factor is x squared. What is its value at x equals 3?

    Answer: ______________

  4. For y prime + 2y = 5, what is the integrating factor?

    • 2x
    • e to the 2x
    • x squared

  5. For y prime + (2 over x) y = 1, the integrating factor is x squared. What is its value at x = 3?

    Answer: ______________

  6. A tank model gives dy/dt + 0.5 y = 2. What value does y settle toward over a long time?

    Answer: ______________

  7. Solve y prime + 2y = e to the x. Which family is the answer?

    • y = e to the x + C
    • y = e to the 2x over 3 + C
    • y = e to the x over 3 + C e to the minus 2x

  8. Multiplying by the integrating factor turns the left side into the derivative of the factor times y.

    Circle one:   True   False

  9. Pat builds the factor straight from 2y prime + 4y = 6 using P = 4. What is wrong?

    • The factor needs Q instead of P
    • Pat must divide first so y prime has coefficient 1
    • The factor needs a minus sign on P

  10. You check y = e to the x over 3 in y prime + 2y. What do you get?

    • e to the x
    • e to the 2x
    • 3 e to the x
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Answer key

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

The factor that builds a product W1-mt_BXmt2pbWp8-s1

  1. True · That product rule collapse is exactly what the factor is built to create.
  2. y prime plus 2y equals 3 · Divide every term by 2: y prime plus 2y equals 3.
  3. 9 · x squared at 3 is 9.
  4. e to the 2x · P is the constant 2, and e to the integral of 2 dx is e to the 2x.
  5. 9 · 3 squared is 9.
  6. 4 · At rest the rate is zero, so 0.5 y = 2.
  7. y = e to the x over 3 + C e to the minus 2x · The factor e to the 2x collapses the left, and one integration finishes it.
  8. True · That product rule collapse is the whole point of the factor.
  9. Pat must divide first so y prime has coefficient 1 · Skipping the division poisons P and everything after it.
  10. e to the x · y prime is e to the x over 3, and adding 2y completes e to the x.
Worksheet · LightMySky