What a Differential Equation Says and What a Solution Is · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Equations about change, functions as answers

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

  1. The general solution is y equals C times e to the x, and y of 0 is 5. What is C?

    Answer: ______________

  2. The function y equals 3 times e to the 2x satisfies the equation y prime equals 2y.

    Circle one:   True   False

  3. A third order equation has a general solution. How many arbitrary constants does it carry?

    • 1
    • 2
    • 3
    • 4

  4. What is the order of the equation y double prime plus 3 y prime plus 2y equals 0?

    • 2
    • 1
    • 3

  5. A third order equation has a general solution. How many arbitrary constants does it carry?

    • 1
    • 2
    • 3

  6. y equals C1 cos x plus C2 sin x satisfies y double prime plus y equals 0, with y of 0 equals 4 and y prime of 0 equals 0. What is C2?

    Answer: ______________

  7. With the same y, y of 0 equals 4 and y prime of 0 equals 0. What is C1?

    Answer: ______________

  8. The general solution is y equals C times e to the x, and y of 0 is 5. What is C?

    Answer: ______________

  9. Pat checks a proposed solution by plugging in x = 1 only. What is wrong?

    • plugging in is the wrong method
    • a solution must work at every point of an interval, not just one
    • derivatives cannot be checked by hand

  10. A second order equation has a general solution with two constants. Adding one initial condition pins down both. Is that right?

    Circle one:   True   False

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Answer key

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

Equations about change, functions as answers W1-mt_tHlHg2nK3o-s1

  1. 5 · At x equals 0, e to the 0 is 1, so C times 1 equals 5.
  2. True · Differentiating gives 6 times e to the 2x, which is exactly 2 times 3 times e to the 2x.
  3. 3 · Each integration introduces one constant, and a third order equation needs three integrations.
  4. 2 · The highest derivative present is the second derivative, so the order is 2.
  5. 3 · Each integration introduces one constant, and a third order equation needs three integrations.
  6. 0 · y prime is minus C1 sin x plus C2 cos x, and at 0 that equals C2, which must be 0.
  7. 4 · y of 0 is C1 times 1 plus C2 times 0, so C1 = 4.
  8. 5 · At x equals 0, e to the 0 is 1, so C times 1 equals 5.
  9. a solution must work at every point of an interval, not just one · Checking means substituting back and confirming both sides match everywhere on an interval.
  10. False · One condition fixes one constant. Order 2 needs two conditions.
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