# Differential Equations: Largest rectangle on ty-plane

1. Feb 11, 2013

### Northbysouth

1. The problem statement, all variables and given/known data
Find the largest open rectangle in the ty-plane that contains the initial value point and satisfies the following theorem:

Let R be the open rectangle defined by a<t<b, α<y<β. Let f(t,y) be a function of two variables defined on R where f(t,y) and the partial derivative ∂f/∂y are continuous on R. Suppose (t0, y0) is a point in R. Then there is an open interval t-interval (c,d) contained in (a, b) and containing t0, in which there exists a unique solution of the initial value problem

y'f(t,y), y(t0) = y0

(t-a)(t-3)y'= 1+ sec(y) and y(π)=1

2. Relevant equations

3. The attempt at a solution

So, when I take the partial derivative I get

∂f(t,y)/∂y = [sec(y)tan(y)](t-1)(t-3)/(t-1)(t-3)2

From this I can see that:

t≠1
t≠3

But y is available for all values.

Hence the largest rectangle goes from 1 to 3 on the t axis and from -∞ to +∞ on the y axis. But this doesn't contain the initial value point. Am i missing something?

2. Feb 12, 2013

### Staff: Mentor

In that case, you need a different rectangle.

3. Feb 12, 2013

### haruspex

I assume that should read y'=f(t,y)
Later you have t-1, not t-a. Which is it?
Why not just sec(y)tan(y)/((t-1)(t-3))?
Think outside the box