- Exercise 2 4 2 Consider A Very General Form Of Newton S Law Of Cooling Given By U T H U T A 2 68 For The Temper 1 (39.79 KiB) Viewed 18 times
- Exercise 2 4 2 Consider A Very General Form Of Newton S Law Of Cooling Given By U T H U T A 2 68 For The Temper 2 (45.99 KiB) Viewed 18 times
(a) Use Theorem 2.4.1 to show that (2.68) has a unique solution for any initial condition u(0) = uo. Hint: the ODE is u' = f(t,u) with f(t,u) =h(u-A). (b) Why would the condition h(0) = 0 make sense to impose on h? Hint: if u(0) = A what should the solution u(t) equal for all t? What is u'(t)? (c) Why would it be reasonable to impose the condition that h must be a decreasing function? Hint: if ui(t) and u2(t) are solutions to u' =h(u - A) with uj(0) > u2(0) > A (object 1 is hotter than object 2), how should u'(0) and u(0) be related? What if Au(0) < u2(0)?
Theorem 2.4.1 – Existence and Uniqueness of Solutions to ODEs. Let R denote a rectangle a<t<b and c<u< d in the tu plane. Suppose that the function f(t,u) is continuous at each point in Rand that a le is continuous at each point in R. Then for any point t = to, u = uo in R, the ODE u'(t) = f(t,u(t)) has a unique solution with u(to) = uo on some interval to - 8 <t<to+d2 with 8 >0 and 2 > 0. =