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Math Help - First order ODE involving substitution

  1. #1
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    First order ODE involving substitution

    Hey guys, I'm new here, rather stuck on this IVP. Full working would be lovely as I haven't come across substitution before and have attempted this problem about 7 times so far..

    By making the substitution z = 3x - y, show that solving the initial value problem

    y' = y -3x -5 -1/(3x -y +8), y(0) = 0,

    leads to

    (3x -y +8)^2 = 65e^2x -1.

    Any help guys? I've tried this problem multiple times, probably doesn't help that I haven't been taught how substitutions come into ODE's.. Many thanks in advance
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  2. #2
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    Re: First order ODE involving substitution

    Using the substitution suggested will make the equation more manageable.

    \displaystyle y' = y-3x-5-\frac{1}{3x -y +8}

    using \displaystyle z=3x-y gives

    \displaystyle y' =-z-5-\frac{1}{z +8}
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  3. #3
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    Re: First order ODE involving substitution

    I see that, but my problem is I can't solve the equation from there, each time I try to solve it I get a different large mess of working that goes nowhere.
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: First order ODE involving substitution

    Quote Originally Posted by pickslides View Post
    Using the substitution suggested will make the equation more manageable.

    \displaystyle y' = y-3x-5-\frac{1}{3x -y +8}

    using \displaystyle z=3x-y gives

    \displaystyle y' =-z-5-\frac{1}{z +8}
    The substitution must be complete and that means that z= 3\ x - y \implies y^{'}= 3 - z^{'} so that the ODE in z is...

    z^{'}= z +8 + \frac{1}{z+8} (1)

    ... where the variables z and x are separated...

    Kind regards

    \chi \sigma
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