Sep 21, 2006 ... (e) Find the exact solution x = ϕ(t) to (1) with the initial condition x(0) = p. ...
discussion and numerical example on page 38 of Strogatz' book.] 3.
Department of Mathematics & Statistics MATH 3043 (1A) ASSIGNMENT # 1
DUE: THURSDAY, SEPTEMBER 21, 2006
1. Consider the differential equation dx = f (x), with f (x) = 1 − x2 , x ∈ R. dt
(1)
(a) Sketch the direction field by hand for −1 ≤ t ≤ 5 and −2 ≤ x ≤ 3. (b) Use the Maple command, dfieldplot, to generate a sketch of the direction field for the same t and x ranges as in part (a). (c) Explain why, on the basis of your results in part (a) or part (b), solutions x = ϕ(t) to (1) with initial value x(0) = p should exist for all t > 0, provided −1 ≤ p < ∞. (d) (i) What are the steady-state solutions to (1)? (ii) Which of these is stable? (e) Find the exact solution x = ϕ(t) to (1) with the initial condition x(0) = p. (f) Use the Maple command, dsolve, to solve (1) for initial condition x(0) = p. 2. Newton’s equation of motion for a falling body is m
dv = mg − kv 2 , ds
(2)
where m, g and k are positive constants, and v(s) is the (downward) velocity at time s. (a) Show that if v and s are re-scaled according to r r mg gk v= · x and s = ·t k m then (2) reduces to (1). (b) Use results from Ex. 1 above to find the terminal velocity v∞ = lim v(s) (when s→∞
it exists). [Note the discussion and numerical example on page 38 of Strogatz’ book.] 3. Ex. 2.3.2 in Strogatz. For part (b), consider three cases: k1 a k1 a (1) x(0) > ; (ii) 0 < x(0) < ; k2 2k2
(iii)
k1 a k1 a < x(0) < 2k2 k2
4. Ex. 2.4.2 in Strogatz. dx = 1 + x2 , x(0) = 0 and determine the dt longest interval −ε < t < ε on which the solution exists (i.e., find ε). (b) Ex. 2.5.2 in Strogatz.
5. (a) Find the exact solution x = ϕ(t) to
6. Ex. 2.5.4 in Strogatz. 7. Ex. 2.7.5 in Strogatz.
