Classifying Second-Order PDEs and What the Type Decides · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Sorting second-order equations by their type

Mathematics · Differential Equations · ages 22-23
Name ______________________   Date ____________

  1. You start a parabolic equation, like the heat equation, with a sharp kink in the data. What happens to the kink as time runs?

    • It smooths out right away
    • It travels along unchanged forever
    • It grows into a bigger and bigger spike

  2. Look at the equation u_xx - 4u_xy + 4u_yy = 0. Using the discriminant B^2 - 4AC, what type is this equation?

    • Elliptic
    • Parabolic
    • Hyperbolic

  3. Look at the equation 2u_xx + 2u_xy + u_yy + 3u_x = 0. What type is it?

    • Elliptic
    • Parabolic
    • Hyperbolic

  4. The heat equation u_t equals 5u_xx has what type?

    • Elliptic
    • Hyperbolic
    • Parabolic

  5. A wave equation is set on a string. Which starting data makes it well posed?

    • The initial position and the initial velocity
    • Only the initial position
    • The temperature at the final time

  6. True or false: the type of a second-order equation (elliptic, parabolic, or hyperbolic) is decided only by the coefficients of its second-order terms, not by the first-order or zero-order terms.

    Circle one:   True   False

  7. Sound in a long pipe obeys u_tt = 4u_xx. Which starting data makes the problem well posed?

    • Only the starting position, with no velocity given
    • The position at some later final time
    • The starting position and the starting velocity

  8. A steady temperature in a flat disc satisfies Laplace's equation. Which data gives a well-posed problem?

    • The temperature and its rate of change on one diameter
    • The temperature at every point of the edge circle
    • The temperature on the top half of the edge only

  9. For 3u_xx plus 2u_xy plus 3u_yy equals 0, compute B squared minus 4AC and name the type.

    • 0, parabolic
    • Negative 32, elliptic
    • 40, hyperbolic

  10. A sharp kink in the starting data of the heat equation is still visible one second later.

    Circle one:   True   False

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Answer key

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Sorting second-order equations by their type W1-mt_hlQWbd1kzR-s1

  1. It smooths out right away · Parabolic equations diffuse: sharp corners in the data are smoothed out instantly, and the solution becomes gentle and rounded.
  2. Parabolic · Here A = 1, B = -4, C = 4, so B^2 - 4AC = 16 - 16 = 0. A zero discriminant means parabolic.
  3. Elliptic · Only the second-order part counts: A = 2, B = 2, C = 1, so B^2 - 4AC = 4 - 8 = -4. Negative means elliptic.
  4. Parabolic · The discriminant is 0, and zero means parabolic.
  5. The initial position and the initial velocity · Second order in time needs two initial conditions: position and velocity.
  6. True · The discriminant B^2 - 4AC uses only A, B, and C, the coefficients of u_xx, u_xy, and u_yy. Lower-order terms do not change the type.
  7. The starting position and the starting velocity · u_tt makes the equation second order in time and hyperbolic, so it needs two pieces of starting data: position and velocity.
  8. The temperature at every point of the edge circle · Laplace's equation is elliptic, so it models a steady state, and that needs the value all around the closed boundary: here the whole edge circle.
  9. Negative 32, elliptic · A 3, B 2, C 3 gives 4 minus 36, which is negative 32: elliptic.
  10. False · Parabolic diffusion smooths every kink right away instead of carrying it.
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