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Math Help - General solution question 2

  1. #1
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    General solution question 2

    Use the method of separating the variables to find the general solution of the equation

    dy/dx - 2xy = 0


    Could somebody please explain this method of separating variables because i have never been taught it (taught different syllabus). Only 4 mark question so guessing its nothing difficult.
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  2. #2
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    Quote Originally Posted by djmccabie View Post
    Use the method of separating the variables to find the general solution of the equation

    dy/dx - 2xy = 0


    Could somebody please explain this method of separating variables because i have never been taught it (taught different syllabus). Only 4 mark question so guessing its nothing difficult.
    dy/dx= 2xy so dy/y= 2xdx. The variables are "separated". Integrate both sides.
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    Thanks guys knew it would be simply
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    would [lny = x^2 + c] be the final answer?
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    Behold, the power of SARDINES!
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    Quote Originally Posted by djmccabie View Post
    would [lny = x^2 + c] be the final answer?
    not quite we need to simplify a bit

    \ln(y)=x^2+c \iff e^{\ln(y)}=e^{x^2+c} \iff y=e^{x^2}e^{c}

    Now we can relabel e^{c}=A just another constant to get

    y=Ae^{x^2}
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    OK i'm struggling to understand that.

    I've remembered that when theres a log in the solution I have been taught to put the constant of integration inside like this:

    [lnAy = x^2] where A is the constant

    so then i get

    e^x = Ay

    y = (1/A)e^x


    however this contradicts your answer
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    Quote Originally Posted by djmccabie View Post
    OK i'm struggling to understand that.

    I've remembered that when theres a log in the solution I have been taught to put the constant of integration inside like this:

    [lnAy = x^2] where A is the constant

    so then i get

    e^x = Ay

    y = (1/A)e^x


    however this contradicts your answer
    No it doesnt. It is an arbitarty constant.

    If you want it can be relabled as

    \frac{1}{A}=B

    The key idea is they are just some unknown constant. If you are given an initial condition then they can be determined
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