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Math Help - Equation

  1. #1
    Senior Member tukeywilliams's Avatar
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    Equation

    The solution of the differential equation for an overdamped vibration has the form x(t) = c_1e^{r_{1}t} + c_2e^{r_{2}t}



    (a) Show that x(t) is 0 at most once

    (b) Show that x'(t) is 0 at most once

    (c) Find the time t when this occurs

    So x(t) = c_1e^{r_{1}t} + c_2e^{r_{2}t}

    What do I do next?

    and x'(t) = c_{1}r_{1}e^{r_{1}t} + c_{2}r_{2}e^{r_{2}t}
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  2. #2
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    Quote Originally Posted by tukeywilliams View Post
    The solution of the differential equation for an overdamped vibration has the form x(t) = c_1e^{r_{1}t} + c_2e^{r_{2}t}



    (a) Show that x(t) is 0 at most once
    Say that c_1,c_2!=0

    We want to solve,
    c_1e^{r_1 t}+c_2e^{r_2 t}=0
    Divide by exp(r_2 t) to get,

    c_1e^{(r_1-r_2)t}+c_2 = 0
    Thus,
    c_1e^{(r_1-r_2)t}=-c_2
    Hence,
    e^{(r_1-r_2)t}=-c_2/c_1
    An exponentaial always have at most one root.
    (b) Show that x'(t) is 0 at most once
    Same idea.
    Differenciating still keeps it an exponential.
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    Senior Member tukeywilliams's Avatar
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