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
and when t=0
am i right about it?
and can somebody give me some hints to deal with the second problem? thank you.