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Thread: Linear , homogenous, separation and change of variables

  1. #46
    Junior Member romeobernard's Avatar
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    Re: Linear , homogenous, separation and change of variables

    (3,-1) is the intersection between two lines right?



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  2. #47
    MHF Contributor MarkFL's Avatar
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    Re: Linear , homogenous, separation and change of variables

    Quote Originally Posted by romeobernard View Post
    (3,-1) is the intersection between two lines right?
    You could look at it that way, we really just needed to solution to the system of linear equations in order to make a good substitution.
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  3. #48
    Junior Member romeobernard's Avatar
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    Re: Linear , homogenous, separation and change of variables

    dv/du = {u+(2v)} / {(2u)+v}

    Next? I cant follow those terms..thanks

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  4. #49
    MHF Contributor MarkFL's Avatar
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    Re: Linear , homogenous, separation and change of variables

    I went further than that and wrote it in first order homogeneous form, which has a well-known theory for solving. You are familiar with such equations, right?
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  5. #50
    Junior Member romeobernard's Avatar
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    Re: Linear , homogenous, separation and change of variables

    Quote Originally Posted by MarkFL View Post
    I went further than that and wrote it in first order homogeneous form, which has a well-known theory for solving. You are familiar with such equations, right?
    Is this right
    V^2(z-1)^3 = c(z+1)

    Turns to this

    (vz-v)^3 = c(vz+v)

    How?thanks

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  6. #51
    Junior Member romeobernard's Avatar
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    Re: Linear , homogenous, separation and change of variables

    Quote Originally Posted by romeobernard View Post
    Is this right
    V^2(z-1)^3 = c(z+1)

    Turns to this

    (vz-v)^3 = c(vz+v)

    How?thanks

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    I saw this problem on the book.

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  7. #52
    MHF Contributor MarkFL's Avatar
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    Re: Linear , homogenous, separation and change of variables

    Well, what I would do is let:

    $\displaystyle w=\frac{v}{u}\implies v=uw\implies \frac{dv}{du}=w+u\frac{dw}{du}$

    And so our ODE becomes:

    $\displaystyle w+u\frac{dw}{du}=\frac{1+2w}{2+w}$

    $\displaystyle u\frac{dw}{du}=\frac{1+2w-w(2+w)}{2+w}$

    $\displaystyle u\frac{dw}{du}=\frac{1-w^2}{2+w}$

    Can you proceed?
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  8. #53
    Junior Member romeobernard's Avatar
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    Re: Linear , homogenous, separation and change of variables

    After those equation (u+2v)du - (2u+v)dv i used the homogeneous DE

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