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Thread: Word Problem Integration

  1. #1
    Senior Member polymerase's Avatar
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    Word Problem Integration

    Let $\displaystyle x$ be a function of $\displaystyle t$ such that $\displaystyle \frac{dx}{dt}=2\sin^2 t\cos^2 x$. Suppose that $\displaystyle x\left(\frac{\pi}{4}\right)=\arctan \left(\frac{1}{2}+\frac{\pi}{4}\right)$. Then $\displaystyle x(0)=$

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  2. #2
    Senior Member Peritus's Avatar
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    $\displaystyle
    \begin{gathered}
    \frac{{dx}}
    {{dt}} = 2\sin ^2 t\cos ^2 x \hfill \\
    \hfill \\
    \Leftrightarrow \frac{1}
    {{\cos ^2 }}dx = 2\sin ^2 tdt \hfill \\
    \end{gathered} $

    $\displaystyle
    \Leftrightarrow \tan x = t - \frac{1}
    {2}\sin 2t + C
    $

    $\displaystyle
    x(t) = a\tan \left( {t - \frac{1}
    {2}\sin 2t} \right) + K
    $

    now apply the I.V. .....
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