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Math Help - homogeneous solution??

  1. #1
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    homogeneous solution??

    Hi guys I am really stuck on this differential equation can anyone help?

    dy/dt = (t^2 + y^2) / (3ty)

    condition - y=1 and t=2

    Thank you in advance
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  2. #2
    MHF Contributor Calculus26's Avatar
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    This is an example of A DE with Homogeneous Coefficients

    ie it can be written in the form dy/dt = f(y/t)
    dy/dt = 1/3 t/y + 1/3 y/t

    Let z= y/t

    tz = y

    and z + tdz/dt = dy/dt

    z + t dz/dt = 1/(3z) + 1/3z

    tdz/dt = 1/(3z) - 2z/3

    you now have a separable equation I'll let you go from here
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  3. #3
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    Could I have solved it by turning it into a seperable equation to begin with??
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  4. #4
    MHF Contributor Calculus26's Avatar
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    No it's not separable--- the problem being the t^2 + y^2 term.

    Typically when fns of y and t are multiplied or divided the eqn can be separated -- when there is addition of fns of y and t you usually cannot separate them.

    Go ahead and try but you'll soon see the eqn won't separate
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  5. #5
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    Ok have tried wont work!

    From where you finished your explanation do I need to integrate each side? I am not to sure how??
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  6. #6
    MHF Contributor Calculus26's Avatar
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    See attachment--you might want to double check my work to see if you agree
    Attached Thumbnails Attached Thumbnails homogeneous solution??-homde.jpg  
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