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Math Help - Second Order Differential Equation

  1. #1
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    Second Order Differential Equation

    Solve [(alpha)^2]u''+a*e^u=0 by multiplying it by du/dx and integrating it in x

    I just keep going in circles and ending up with the original problem.
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  2. #2
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    Re: Second Order Differential Equation

    Quote Originally Posted by cheesecake91 View Post
    Solve [(alpha)^2]u''+a*e^u=0 by multiplying it by du/dx and integrating it in x

    I just keep going in circles and ending up with the original problem.
    \displaystyle \begin{align*} \alpha ^2 \, \frac{d^2u}{dx^2} + a\, e^u &= 0 \\ \alpha ^2 \, \frac{d^2u}{dx^2} &= -a\, e^u \\ \alpha ^2 \, \frac{du}{dx} \, \frac{d^2u}{dx^2} &= -a\, e^u \, \frac{du}{dx}  \end{align*}

    Now, if we let \displaystyle \begin{align*} w = \frac{du}{dx} \end{align*}, then \displaystyle \begin{align*} \frac{dw}{dx} = \frac{d^2u}{dx^2} \end{align*} and the DE becomes

    \displaystyle \begin{align*} \alpha ^2 \, w \, \frac{dw}{dx} &= -a \, e^u \, \frac{du}{dx} \\ \int{ \alpha^2 \, w \, \frac{dw}{dx} \, dx} &= \int{ -a\, e^u \, \frac{du}{dx} \, dx} \\ \int{\alpha^2 \, w \, dw} &= \int{ -a \, e^u \, du} \end{align*}

    Finish it.
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  3. #3
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    Re: Second Order Differential Equation

    Thanks! I was thinking it had something to do with substitution but couldn't figure it out.
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