Question

Suppose that f(t) is the unique solution to the IVP y' = t + y^2 , y(0) = 5 and g(t) is the unique solution to the IVP y' = 1/(y + t^2) , y(5) = 0.

a. Determine an IVP that the function y = f(g(t)) solves. [Hint: You differential equation part will contain the functions t, g(t), and y in its expression.

b. (2 points) Show that the function y = t also solves this IVP.

c. (2 points) Use the uniqueness part of the Existence & Uniqueness Theorem to conclude that f and g are inverses.

Answer 1

a) If then by chain rule. Since satisfies we have , and similarly,

Hence,

Thus, the IVP is

b) If then and

and . Thus, solves

c) The function

is Lipschitz in . By uniqueness of solution to IVP

we get that . Thus, are inverses.