MTH 343 · 5.1 The elementary theory of initial-value problems

Initial-Value Problem Lab

Choose an initial-value problem y′ = f(t, y), a ≤ t ≤ b, y(a) = α from Exercise Set 5.1 or the slides. The lab shows whether the hypotheses of Theorem 5.4 hold, with the reason and the Lipschitz constant, draws the slope field with the solution, and checks the solution you found by hand.

Exercise Set 5.1 (9th edition)

Exercise 3 uses 0 ≤ t ≤ 1 and y(0) = 1; its part (ii) asks about Theorem 5.6, which has the same hypotheses as Theorem 5.4.

From the slides
Selected problem
y(t) =

Type the y(t) you found by hand, using t only: + − * / ^, sin, cos, exp, ln, sqrt, cbrt, abs, pi, e, and lambertW(x) for 2(d). The lab checks y(a) = α and y′(t) = f(t, y(t)) on [a, b]. Choosing a problem clears this box (three slide examples fill in a solution to try).

1. f continuous on D? —
2. Lipschitz condition in y on D? —
3. Theorem 5.4 —
Your solution —

Slope field and solution

    How large does |∂f/∂y| get on D?

      For each y, the curve shows the largest |∂f/∂y(t, y)| over a ≤ t ≤ b, from the exact partial derivative. If D is convex and the curve stays below a horizontal line L, Theorem 5.3 gives a Lipschitz condition with constant L. The vertical scale is logarithmic; on the strip the horizontal y scale is also stretched so that values from −10⁶ to 10⁶ fit.

      How to show an IVP has a unique solution

      The five steps from the slides, worked out for the selected problem.

        Values on [a, b]