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Math Help - Formula that gives all possible solutions

  1. #1
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    Formula that gives all possible solutions

    Hi there,
    I'm working on a problem that asks me to find a formula that'll give all possible solutions to the equation V'' + (2/3)V'(r) = 0. I'm not certain where to begin. I believe that solving for V'(r) would be a nice first step, but I don't know how to go about this either. ANy guidance would be greatly appreciated. Thank you!
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    Re: Formula that gives all possible solutions

    Look at this link under "Second Order Differential equations", the first example will show you how to proceed.

    First and Second Order Differential Equations
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    Re: Formula that gives all possible solutions

    Thank you for the link pickslides.

    From what I've gotten, the solution should be something akin to y = c(sub 1)e^(r(sub1)x) + c(sub 2)e^(r(sub 2)x). The solution to r^2 - (2/r)r = 0 is plus or minus square root of 2. The solution, then, appears to be y = c(sub 1)e^(sqrt(2)x) + c(sub 2)e^(-sqrt(2)x). Is this correct? Thank you once again for your assistance.
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    Re: Formula that gives all possible solutions

    \displaystyle \lambda^2+\frac{2}{3}\lambda = 0

    \displaystyle \lambda\left( \lambda+\frac{2}{3}\right) = 0

    By the null factor law \lambda = 0, \frac{-2}{3}

    Now the solution to V(r)

    Now \displaystyle V(r) = C_1e^{\frac{-2r}{3}}+C_2e^{0}
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    Re: Formula that gives all possible solutions

    pickslides, I'm so sorry. I typed the coefficient (2/3) in the original problem when the problem states (2/r). I typed my attempt at a solution with regard to that. I apologize sincerely for my mistake. You've been very kind with your explanation, and I thank you.
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  6. #6
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    Re: Formula that gives all possible solutions

    This makes things more difficult.

    Have a look at this link

    solve y''+(2/x)y' =0 - Wolfram|Alpha
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    Re: Formula that gives all possible solutions

    Quote Originally Posted by reliance View Post
    Hi there,
    I'm working on a problem that asks me to find a formula that'll give all possible solutions to the equation V'' + (2/3)V'(r) = 0. I'm not certain where to begin. I believe that solving for V'(r) would be a nice first step, but I don't know how to go about this either. ANy guidance would be greatly appreciated. Thank you!
    The equation is V''+ (2/r)V'= 0. Since V itself does not appear in the equation, let U= V' so the equation becomes U'+ (2/r)U= 0. That is a separable first order equation. We can write dU/dr= -(2/r)U and then dU/U= -(2 dr/r). Integrating both sides, ln(U)= -2ln(r)+ C= ln(r^{-2}+ C so that U= C' r^{-2} where C'= e^C. Going back to V we have \frac{dV}{dr}= C'r^{-2} and that's an easy integral.
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