Existence and Uniqueness Theorem (Help)

**Markeur** · November 12th 2010, 03:40 AM

Let $f(t,x)=1-x.sin(t)$ , for $(t,x) \in \mathbb{R}^2$ .
(a)Show that $f$ satisfies a Lipschitz condition on $\mathbb{R}^2$ with respect to $x$ . (Solved)
(b)Find the first 3 successive approximations $x_0(t)$ , $x_1(t)$ and $x_2(t)$ of the initial value problem $x^{'}(t)=f(t,x)$ , $x(0)=0$ . (Solved)
(c)State the largest interval $I$ in which a unique solution to the initial value problem in (b) is defined. (Solved)
(d)Let $\phi(t)$ be the solution to the initial value problem in (b). Prove that $\vert \phi(t) \vert \leq e^2 \vert t \vert}$ , for all $t \in I$ .

I've difficulty in doing the last questions. How do I go about doing the last question?

Thanks in advance.

**Ackbeet** · November 12th 2010, 04:46 AM

A couple of questions/comments:

1. What was your answer to (c)?
2. Why not attempt a regular solution of the DE? It's first-order linear, right? You could at least get some sort of formula. What do you get?

**Markeur** · November 12th 2010, 05:24 AM

1. My answer to (c) is $\mathbb{R}$
2. I got $x=e^{cos(t)} \int e^{-cos(t)} dt$

So how do I continue from here?

**Ackbeet** · November 12th 2010, 05:28 AM

Here's where you get to play around with inequalities. Here are a few that might come in handy:

$\displaystyle\left|\int f(t)\,dt\right|\le\int|f(t)|\,dt.$

$|\cos(t)|\le 1.$

If $a<b,$ then because the exponential function is monotone increasing, it follows that $e^{a}<e^{b}.$

I think that might be about all you need. So what could you do next?

**Markeur** · November 12th 2010, 05:58 AM

$\vert \phi(t) \vert = \vert e^{cos(t)} \int_{0}^{t} e^{-cos(s)} ds \vert = \vert e^{cos(t)} \vert \vert \int_{0}^{t} e^{-cos(s)} ds \vert \leq e \vert \int_{0}^{t} \vert e^{-cos(s)} \vert ds \vert \leq e \vert \int_{0}^{t} e \; ds \vert = e^{2}\vert t \vert$

Alright I think I got it. Thanks.

**Ackbeet** · November 12th 2010, 06:00 AM

Looks fine to me. You're welcome, and have a good one!

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