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

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?

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

  2. What is the characteristic equation of y double prime minus 3 y prime plus 2y equals 0?

    • r squared minus 3r plus 2 equals 0
    • r squared plus 3r plus 2 equals 0
    • r squared minus 3r equals 0
    • 2r squared minus 3r plus 1 equals 0

  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 plus C2 times x) times e to the 2x
    • 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 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 pure polynomial
    • Two distinct real exponentials
    • An oscillation wrapped in an exponential

  8. Solve y double prime plus 4 y prime plus 4 y equals 0. What is the general solution?

    • C1 e to the minus 2 x plus C2 e to the 2 x
    • C1 plus C2 times x, all times e to the minus 2 x
    • C1 cosine 2 x plus C2 sine 2 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 stays free, so the answer is never pinned down
    • 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. r squared minus 3r plus 2 equals 0 · Each derivative becomes a power of r with the same coefficients: r squared minus 3r plus 2.
  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 oscillation wrapped in an exponential · Complex roots rewrite as sine and cosine times exponential growth or decay.
  8. C1 plus C2 times x, all times e to the minus 2 x · The root minus 2 repeats, so the second piece carries the x factor.
  9. One constant stays free, so the answer is never pinned down · Two constants need two conditions: position plus velocity together.
  10. The repeated root 3 · The lone x factor on the second piece is the signature of repetition.
Worksheet · LightMySky