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Math Help - Determine r for Explicit Formula

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    Determine r for Explicit Formula

    Determine what the value(s) of r has to be in order to find explicit formulas (that is, formulas without integrals) for solutions to \displaystyle \frac{dy}{dt}=t^ry+4. Determine what the general solutions are for each value of r.
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  2. #2
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    Just by looking at it, -1\leq{r}\leq{0} have a few good ones.

    Do you see why, say 2, will not work?.

    If r=2, then you have an integrating factor of t^{2}.

    Then e^{\int{t^{2}}dt}=e^{\frac{t^{3}}{3}}

    Which is not integrable by elementary means.

    Actually, e^{\int{t^{r}}}dt=e^{\frac{t^{r+1}}{r+1}} are generally not easily integrated.
    Trying to integrate \int{e^{\frac{t^{r+1}}{r+1}}}dt can be a booger.

    Then you have for r=0 e^{\int{dt}}=e^{t}

    If r=-1, then e^{\int\frac{1}{t}dt}=t

    And so on. Just something to think about. See if you can find others in the interval that will or will not work.
    Last edited by galactus; October 13th 2007 at 04:03 PM.
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