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
<br /> y&#039;&#039;-\frac{1}{y}(y&#039;)^{2}=0\Rightarrow\frac{y&#039;&#039;}{y&#039;}=\frac{y&#039;}{y}<br />

 

[dextercioby]

<br /> y&#039;&#039;-\frac{1}{y}(y&#039;)^{2}=0\Rightarrow yy&#039;&#039;= y&#039;^{2}

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