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.
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Equations about change, functions as answers

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

  1. For the proposed function y=x²+1 and equation y'=y, calculate the residual y'-y at x=2.

    Answer: ______________

  2. The function y=x²+2 solves the initial-value problem y'=2x, y(0)=3.

    Circle one:   True   False

  3. For y''=2, the general solution is y=x²+Ax+B. How many independent arbitrary constants does this formula contain?

    • 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. How many independent arbitrary constants are in the general solution of y'''=0?

    • 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. Let y=C1 cos x+C2 sin x, with y(0)=4 and y'(0)=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 substitutes a proposed function into a differential equation only at x=1. Why is this check not enough?

    • Substitution cannot test equations containing derivatives
    • The equation must hold throughout a solution interval
    • Derivatives can only be evaluated at the initial input

  10. For y''=0, the general solution is y=Ax+B. The single condition y(0)=4 determines both A and B.

    Circle one:   True   False

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Answer key

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

  1. -1 · At x=2, y'=4 and y=5. The residual is 4-5=-1, so the function does not solve the equation on an interval containing 2.
  2. False · Its derivative is 2x, but y(0)=2 rather than 3. It satisfies the equation without satisfying the stated initial value.
  3. 2 · A and B can vary. The coefficient of x² is fixed at 1, so it is not an arbitrary constant.
  4. 2 · The highest derivative present is the second derivative, so the order is 2.
  5. 3 · Three integrations give y''=K1, y'=K1x+K2 and y=K1x²/2+K2x+K3.
  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. The equation must hold throughout a solution interval · Equality at x=1 checks one point. A solution must satisfy the differential equation at every point of an interval.
  10. False · Substitution gives B=4. A remains arbitrary, so many functions in the family meet this condition.
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