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Solving 2 differential equations

  1. Mar 31, 2012 #1
    dx/dt=ay and dy/dt=bx where x and y are function of t [x(t) and y(t)] and a and b are constant.

    1) show what x and y satisfy the equation for a hyperbola: y^2-(b/a)*x^2=(y_0)^2-(b/a)*(x_0)^2

    2) suppose at some time t_s, the point (x(t_s),y(t_s)) lies on the upper branch of hyperbola, show that: y(t_s)>sqrt(b/a)*x(t_s)

    I dun know whether i am doing it right.

    First, in integrate both equations,

    dx/dt=ay >>> x/y+C_1=at+C_2 >>> x/y+C_5=at

    dy/dt=bx >>> y/x+C_3=bt+C_4 >>> y/x+C_6=bt

    then I say t = 0 and so

    x/y+C_5=at >>> C_5=-x_0/y_0

    y/x+C_6=bt >>> C_6=-y_0/x_0

    then i say this happens only when C_5 and C_6 are 0

    then going back to

    x/y+C_5=at >>> x/y=at

    y/x+C_6=bt >>> y/x=bt and isolating t to yield

    y^2-(b/a)*x^2=0

    and when t=0

    y_0^2-(b/a)*x_0^2=0

    so y^2-(b/a)*x^2=y_0^2-(b/a)*x_0^2

    am i right about it?

    and can somebody give me some hints to deal with the second problem? thank you.
     
  2. jcsd
  3. Mar 31, 2012 #2

    tiny-tim

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    hi oxxiissiixxo! :smile:

    (try using the X2 button just above the Reply box :wink:)
    i've no idea what you're doing here :redface:

    you can't possibly integrate those equations​

    hint: try differentiating the equation they've given you :wink:
     
  4. Mar 31, 2012 #3

    Dick

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    You can't integrate the equations like that. dx/y isn't d(x/y). I suggest you differentiate the hyperbola equation.
     
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