Series Solutions About an Ordinary Point · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Guessing a power series for the answer

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

  1. y' = -y with y = Σ a_n x^n and a_0 = 4. The recurrence is (n+1) a_{n+1} = -a_n. What is a_2?

    Answer: ______________

  2. For y prime equals 2 y with a0 equals 5, the series recurrence is n plus 1 times a sub n plus 1 equals 2 a sub n. Type a1.

    Answer: ______________

  3. For y prime equals 5 y in series form, which recurrence is correct?

    • n times a sub n equals 5 a sub n plus 1
    • a sub n plus 1 equals 5 times n plus 1 times a sub n
    • n plus 1 times a sub n plus 1 equals 5 a sub n

  4. For y prime equals 2 y the series recurrence is n plus 1 times a sub n plus 1 equals 2 a sub n. If a1 is 10, what is a2?

    Answer: ______________

  5. The radius of convergence tells you how far from the centre the series solution stays valid.

    Circle one:   True   False

  6. y'' = y with y(0) = 1, y'(0) = 0 gives an even series with a_0 = 1. What is a_2?

    Answer: ______________

  7. y'' = 4y with y(0) = 3 and y'(0) = 0. The recurrence is (n+2)(n+1) a_{n+2} = 4 a_n. What is a_2?

    Answer: ______________

  8. Why shift indices before matching coefficients?

    • So every sum speaks in the same power of x
    • To change the radius of convergence of the series
    • To remove the initial conditions from the problem

  9. A student matches coefficients without shifting indices and gets a wrong recurrence. What broke?

    • Nothing, shifting is optional decoration
    • Unlike powers were equated
    • The initial conditions were applied twice

  10. For y double prime minus x y equals 0 about x equals 0, what do the initial conditions determine?

    • The radius of convergence of the whole series
    • The starting coefficients a0 and a1
    • Whether the point is ordinary
LightMySky · lightmysky.comW1-mt_E6NiSd7MED-s1

Answer key

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

Guessing a power series for the answer W1-mt_E6NiSd7MED-s1

  1. 2 · a_1 = -4, then 2a_2 = -a_1 = 4, so a_2 = 2.
  2. 10 · At n equals 0 the recurrence reads 1 times a1 equals 2 times 5.
  3. n plus 1 times a sub n plus 1 equals 5 a sub n · Differentiating shifts the index, leaving n plus 1 on the new coefficient, and the right side is 5 a sub n.
  4. 10 · At n equals 1 the recurrence reads 2 times a2 equals 2 times 10.
  5. True · Beyond that distance the sum may diverge and stop representing the solution.
  6. 0.5 · Matching x^0: 2·a_2 = a_0 = 1, so a_2 = 0.5.
  7. 6 · a_0 = y(0) = 3. At n = 0: 2·1·a_2 = 4·3 = 12, so a_2 = 6.
  8. So every sum speaks in the same power of x · Coefficients can only be compared once like powers sit side by side.
  9. Unlike powers were equated · Unshifted sums pair x to the n with x to the n plus 1, scrambling the relation.
  10. The starting coefficients a0 and a1 · Seed values like a0 and a1 generate every later coefficient through the recurrence.
Worksheet · LightMySky