How can I solve this second order differential equation?

[raul_l]

Homework Statement

dx/dt=1-1/y
dy/dt=1/(x-t)

The Attempt at a Solution

If I take the derivative of the second equation and substitute it to the first one I

$$\frac{d^2 y}{dt^2} - \frac{1}{y} (\frac{dy}{dt})^2 = 0$$

but I don't know how to solve this. Could anyone name any methods that could try?
Thanks.

 

[hunt_mat]

The trick is to notice
$$y''-\frac{1}{y}(y')^{2}=0\Rightarrow\frac{y''}{y'}=\frac{y'}{y}$$

 

[dextercioby]

$$y''-\frac{1}{y}(y')^{2}=0\Rightarrow yy''= y'^{2}$$

Now put in the equation $y=Ae^{Bt}$ and find A and B.