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Thread: Existence and Uniqueness Theorem (Help)

  1. #1
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    Existence and Uniqueness Theorem (Help)

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

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

    Thanks in advance.
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  2. #2
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    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?
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  3. #3
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    1. My answer to (c) is $\displaystyle \mathbb{R}$
    2. I got $\displaystyle x=e^{cos(t)} \int e^{-cos(t)} dt$

    So how do I continue from here?
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  4. #4
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    Here's where you get to play around with inequalities. Here are a few that might come in handy:

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

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

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

    I think that might be about all you need. So what could you do next?
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  5. #5
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    $\displaystyle \vert \phi(t) \vert = \vert e^{cos(t)} \int_{0}^{t} e^{-cos(s)} ds \vert = \vert e^{cos(t)} \vert \vert \int_{0}^{t} e^{-cos(s)} ds \vert \leq e \vert \int_{0}^{t} \vert e^{-cos(s)} \vert ds \vert \leq e \vert \int_{0}^{t} e \; ds \vert = e^{2}\vert t \vert$

    Alright I think I got it. Thanks.
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  6. #6
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    Looks fine to me. You're welcome, and have a good one!
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