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Math Help - differential equation

  1. #1
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    differential equation

    I am quite confused with the following question. Would appreciate any help to get me started. Thanks
    Find one set of values of the numbers a and b, such that the function y = e^ax + bln(x) is a solution of the differential equations given below.

    y ay = 0
    b + y b/x = 0
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  2. #2
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    Quote Originally Posted by mellowdano View Post
    I am quite confused with the following question. Would appreciate any help to get me started. Thanks
    Find one set of values of the numbers a and b, such that the function y = e^ax + bln(x) is a solution of the differential equations given below.

    y ay = 0
    b + y b/x = 0
    You are told that y = e^{ax} + b\ln{x}.

    So y' = ae^{ax} + \frac{b}{x}

    y'' = a^2e^{ax} - \frac{b}{x^2}.


    So equation 1:

    y'' - ay' = 0

    a^2e^{ax} - \frac{b}{x^2} - a\left(ae^{ax} + \frac{b}{x}\right) = 0

    a^2e^{ax} - \frac{b}{x^2} - a^2e^{ax} + \frac{ab}{x} = 0

    \frac{abx - b}{x} = 0

    abx - b = 0

    b(ax - 1) = 0


    This means either b = 0 or a = \frac{1}{x}.


    Equation 2:

    b' + y - \frac{b}{x} = 0...

    Are you sure you copied this down correctly?
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  3. #3
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    Thanks! And yes the second one has a mistake it should be
    y' + y - b/x = 0

    So i'm thinking for equation 2;
    ae^(ax) + b/x + e^(ax) + b ln(x) b/x = 0
    ae^(ax) + e^(ax) + b ln(x) = 0
    a^2 ln(x) + a ln(x) + b ln(x) = 0
    a^2 + a + b = 0
    a(a +1 + b) = 0
    a = 0
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