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Math Help - [SOLVED] Another Separable ODE

  1. #1
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    [SOLVED] Another Separable ODE

    Ok another one

    dy/dx = 2y/1- x^4

    sol: y = C|1+x/1-x|^1/2 e^arctanx


    Very confusing this is how i start out:

    => |2y dy = | 1/1-x^4 dx

    => y^2 = 1/4 lnu

    => y^2 = 1/4 ln(1-x^4) +c

    => y = (1/4 ln(1-x^4)+c )^1/2

    then im lost, and ive probably stuffed it up right from near the start. im not sure if i should be using integration by substitution when integrating the x side.
    Help appreciated
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  2. #2
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    Quote Originally Posted by sterps View Post
    Ok another one

    dy/dx = 2y/1- x^4

    sol: y = C|1+x/1-x|^1/2 e^arctanx


    Very confusing this is how i start out:

    => |2y dy = | 1/1-x^4 dx

    => y^2 = 1/4 lnu

    => y^2 = 1/4 ln(1-x^4) +c

    => y = (1/4 ln(1-x^4)+c )^1/2

    then im lost, and ive probably stuffed it up right from near the start. im not sure if i should be using integration by substitution when integrating the x side.
    Help appreciated
    You've made a number of errors. Note: \int \frac{dy}{2y} = \int \frac{dx}{1 - x^4}.

    I suggest partial fractions for the integral on the right hand side. Use the decomposition \frac{1}{1 - x^4} = \frac{A}{1 - x} + \frac{B}{1 + x} + \frac{Cx + D}{1 + x^2}.
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  3. #3
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    Quote Originally Posted by sterps View Post

    => |2y dy = | 1/1-x^4 dx

    => y^2 = 1/4 lnu

    => y^2 = 1/4 ln(1-x^4) +c

    => y = (1/4 ln(1-x^4)+c )^1/2
    your integration is wrong
    1/4 ln(1-x^4) +c is not anti-derivative of 1/1-x^4

    Do you know partial fraction?

    EDIT : wew, a little bit late. Mr_fantastic has given the clue ^^
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  4. #4
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    Quote Originally Posted by sterps View Post
    Ok another one

    dy/dx = 2y/1- x^4

    If you are going to post you question in plain ASCII use sufficient brackets to render you meaning unambiguous:

    dy/dx = 2y/(1- x^4)

    would do here.

    CB
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  5. #5
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    \frac{1}{1-x^{4}}=\frac{1}{\left( 1-x^{2} \right)\left( 1+x^{2} \right)}=\frac{1}{2}\cdot \frac{\left( 1+x^{2} \right)+\left( 1-x^{2} \right)}{\left( 1-x^{2} \right)\left( 1+x^{2} \right)}.
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