No, this is not at all correct. You have integrated the first equationassumingthat y is a constant and integrating the second equationassumingthat x is constant and those assumptions are not true.

There are two standard ways to handling pairs of equations.

Simplest is to [b]differentiate[b] the first equation as second time with respect to t: d^2x/dt^2= a(dy/dt) and, from the second equation that becomes

d^2x/dt^2= a(bx) so you have the second degree equation d^2x/dt^2- abx= 0. The general solution to that will involve two undetermined constants. Once you have x, solve for y by using y= (dx/dt)/a.

\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.