Second-Order Linear Equations with Constant Coefficients · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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One guess that cracks constant equations

Mathematics · Differential Equations · ages 20-21
Name ______________________   Date ____________

  1. The roots are 1 plus 2i and 1 minus 2i. What is the general solution?

    • (C1 plus C2 x) times e to the x
    • C1 e to the x plus C2 e to the minus x
    • C1 cos 2x plus C2 sin 2x
    • e to the x times (C1 cos 2x plus C2 sin 2x)

  2. What is the characteristic equation of 2y'' + 5y' - 3y = 0?

    • r squared plus 5r minus 3 equals 0
    • 2 r squared plus 5r minus 3 equals 0
    • 2 r squared minus 5r plus 3 equals 0
    • 2r plus 5 equals 3

  3. The pair minus 1 plus or minus 3i gives decaying oscillations e to the minus x times (C1 cos 3x plus C2 sin 3x). True or false?

    Circle one:   True   False

  4. The characteristic equation has the repeated root r equals 2. What is the general solution?

    • C1 e to the 2x plus C2 e to the minus 2x
    • e to the 2x times (C1 cos x plus C2 sin x)
    • (C1 plus C2 times x) times e to the 2x
    • C1 plus C2 times e to the 2x

  5. Solve r squared minus 3 r plus 2 equals 0. Type the larger root.

    Answer: ______________

  6. For a repeated root, C1 e to the r x plus C2 e to the r x is already the general solution.

    Circle one:   True   False

  7. The characteristic roots are a complex pair. What does the solution look like?

    • A polynomial times one real exponential
    • Two different real exponentials added
    • An exponential times sine and cosine

  8. Solve y double prime plus 6 y prime plus 9 y equals 0. What is the general solution?

    • C1 e to the minus 3 x plus C2 e to the 3 x
    • (C1 plus C2 x) e to the minus 3 x
    • C1 cosine 3 x plus C2 sine 3 x

  9. A student solves a second order problem using only the starting position, ignoring velocity. What goes wrong?

    • Nothing, one condition always suffices
    • One constant is left free and unknown
    • The characteristic equation changes

  10. Someone hands you C1 e to the 3 x plus C2 times x e to the 3 x as a general solution. What root case produced it?

    • The repeated root 3
    • Distinct roots 3 and 1
    • A complex pair around 3
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Answer key

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

One guess that cracks constant equations W1-mt_G7sMq1EHwQ-s1

  1. e to the x times (C1 cos 2x plus C2 sin 2x) · The real part 1 goes in the exponential and the imaginary part 2 goes inside sine and cosine.
  2. 2 r squared plus 5r minus 3 equals 0 · Each derivative becomes a power of r with the same coefficients: 2r² + 5r - 3.
  3. True · Negative real part means the exponential decays while the imaginary part drives the oscillation.
  4. (C1 plus C2 times x) times e to the 2x · A double root gives e to the rx and x times e to the rx as the independent pair.
  5. 2 · Factoring gives r minus 1 times r minus 2, so the roots are 1 and 2.
  6. False · Those two pieces are the same function, so the x factor is needed for independence.
  7. An exponential times sine and cosine · Complex roots a ± bi give e to the ax times (C1 cos bx + C2 sin bx): sine and cosine inside an exponential.
  8. (C1 plus C2 x) e to the minus 3 x · The root minus 3 repeats, so the second piece carries the x factor.
  9. One constant is left free and unknown · Two constants need two conditions: position plus velocity together.
  10. The repeated root 3 · The x factor on the second piece marks a repeated root.
Worksheet · LightMySky