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Thread: Ordinary equations help

  1. #1
    Newbie kirbyiwaki's Avatar
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    Ordinary equations help

    It happens that I'm not good with integrals, and our teacher is a good teacher, but has a little problem: he uses easy examples but gives us difficult homework... I've managed to do most of it, but have trouble with last ones... can anyone help me please? I'm desperate !!!

    I-Solve the next lineal equations:

    4.-

    II- Find the general solution for the next exact ordinary equations:



    (I know the first one of these is not exact... but can it be solved with an integrating factor?... and the last one... I know it says general solution too... do you think is a trap from my teacher [last time he said "one of the problems didn't have any solution", so I get that wrong...])

    Also... procedures will be highly appreciated n__n"

    And, after the answers... can someone teach me the art of EDO?... I'll be very grateful, thanks in advance.
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  2. #2
    Senior Member Peritus's Avatar
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    $\displaystyle
    \begin{gathered}
    \frac{{dP}}
    {{dt}} + 2tP = P + 4t - 2 \hfill \\
    \Leftrightarrow \frac{{dP}}
    {{dt}} + (2t - 1)P = 4t - 2 \hfill \\
    \Leftrightarrow e^{t^2 - t} \frac{{dP}}
    {{dt}} + e^{t^2 - t} (2t - 1)P = e^{t^2 - t} \left( {4t - 2} \right) \hfill \\
    \end{gathered}
    $

    $\displaystyle
    \begin{gathered}
    \Leftrightarrow \frac{d}
    {{dt}}\left( {e^{t^2 - t} P} \right) = e^{t^2 - t} \left( {4t - 2} \right) \hfill \\
    \Leftrightarrow e^{t^2 - t} P = \int {e^{t^2 - t} \left( {4t - 2} \right)dt = 2} \int {\left( {2t - 1} \right)e^{t^2 - t} dt = 2} e^{t^2 - t} + C \hfill \\
    \Leftrightarrow P(t) = 2 + Ce^{t - t^2 } \hfill \\
    \end{gathered}
    $

    Exact First-Order Ordinary Differential Equation -- from Wolfram MathWorld
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  3. #3
    Newbie kirbyiwaki's Avatar
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    Thank you~!!! I had it almost done (via general formula, tough), but the integration ruined me u.u... thanks again ^^!
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