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Math Help - differential equation to algebraic form

  1. #1
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    differential equation to algebraic form

    using substitution x = e^(kt) convert the following differential equation into an algebraic equation for k (d^2)x/dt^2 + 2B(dx/dt) + (w^2)x = 0 with w>0 after substituting for x what must i do to convert to algebraic form i was not taught anything like this in lectures
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  2. #2
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    Hello, crafty!

    This is one of the classic methods for Differential Equations.
    It should have been demonstrated to you in a lecture.


    Using substitution x = e^{kt}, convert the following differential equation
    into an algebraic equation for k\!:\;\;\frac{d^2x}{dt^2} + 2B\frac{dx}{dt} + w^2x \:= \:0\;\;\text{with }w>0

    After substituting for x, what must i do to convert to algebraic form?
    i was not taught anything like this in lectures.
    We have: . \frac{d^2x}{dt^2} + 2B\frac{dx}{dt} + w^2x \:=\:0 .[1]

    . . And: . \begin{Bmatrix} x &=& e^{kt} \\ \\[-4mm]\dfrac{dx}{dt} &=& ke^{kt} \\ \\[-4mm] \dfrac{d^2x}{dt^2} &=& k^2e^{kt} \end{Bmatrix}


    Substitute into [1]: . k^2e^{kt} + 2B\!\cdot\!ke^{kt} + w^2\!\cdot\!e^{kt} \;=\;0


    Divide by e^{kt}\!:\;\;k^2 + 2Bk + w^2 \;=\;0 . (This is the "algebraic form".)

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  3. #3
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    thanks
    Last edited by crafty; February 5th 2009 at 12:33 PM.
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