Show that is a solution of the initial value problem , where y(0) = 0, in the...

Show that

is a solution of the initial value problem , where y(0) = 0, in the sense of Definition 1.15 from Section 2.1. Find a second solution and explain why this lack of uniqueness does not contradict Theorem 7.16.

Definition 1.15:

A first order differential equation together with an initial condition,

Is called an initial value problem. A solution of the initial value problem is a differentiable function y(t) such that

1. for all t in an interval containing where y (t) is defined, and

2.

THEOREM 7.16 Uniqueness of solutions

Suppose the function f (t, x) and its partial derivative ∂f/∂x are both continuous on the rectangle R in the tx-plane. Suppose and that the solutions

Then as long as (t, x(t)) and (t, y(t)) stay in R, we have

x(t) = y(t).