Question

1. Consider the differential equation dy/dt = 2?square root(absolute value of y) with initial condition y(t0)=y0

• For what values of y0 does the Existence Theorem apply?
• For what values of y0 does the Uniqueness theorem apply?
• Verify that y1(t) = 0 solves the initial value problem with y0 = 0
• Verify that y2(t) = t2 solves the initial value problem with y0 = 0
• Does this violate the theorems from this section 1.5? Why or why not?

Answer 1

The general first-order ODE is
  
We are interested in the following questions:
(i) Under what conditions can we be sure that a solution to exists?
(ii) Under what conditions can we be sure that there is
a unique solution to Here is the answers.

Theorem 1 (Existence). Suppose that is a continuous function defined in some region

containing the point Then there exists a number so that a solution
to   is defined for .

Theorem 2 (Uniqueness). Suppose that both and are continuous functions defined on a region as in Theorem 1, Then there exists a number (possibly smaller than ) so that the solution to whose existence was guaranteed by Theorem 1, is the unique solution to for

Here

1. Since is continuous everywhere in , By Theorem 1, we have the existence of the solution is guaranteed for any real value

2. Since is continuous everywhere in , By Theorem 2, we have the uniqueness of the solution is guaranteed for any real value

3.