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Math Help - First Order Homogenous Differential Equation

  1. #1
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    First Order Homogenous Differential Equation

    By using the substitution y=vx, show that the general solution of the first order homogeneous differential equation (x+y)[dy/dx]= y-x in the case where x>0 is given by y=k, where k is a constant.

    Many thanks in advance!
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  2. #2
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    Quote Originally Posted by cyt91 View Post
    By using the substitution y=vx, show that the general solution of the first order homogeneous differential equation (x+y)[dy/dx]= y-x in the case where x>0 is given by y=k, where k is a constant.

    Many thanks in advance!
    Okay, have you tried this at all? You can write the given equation as (x+y)dy= (y- x)dx. If y= vx, then dy= v dx+ x dv. Replace dy with that and replace y with xv and see what you get!

    Far better to do it yourself than have someone do it for you.
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  3. #3
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    Yes, I did. I came to

    arc tan [y/x] + ln {[(y^2 + x^2)^0.5]/[x^2]} = ln {A/x}

    where A is a constant.

    Now, how do you obtain the expression y=k where k is a constant?

    Is my solution correct?

    Thanks a lot!
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  4. #4
    MHF Contributor Calculus26's Avatar
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    Check your question because y = K is not the general solution

    If y = K were the solution

    dy/dx = 0 = (K - x)/(K+x) is only true for the single value x= K

    you're solution matches what i got
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  5. #5
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    I've checked the question with my instructor. Yea, the question has problems. It's the book's publisher's mistake. Thanks anyway. Good thing I got the general solution right.
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