Find the integral curves then the trajectory of the vector field associated to the differential system $$\begin{cases} x'(t)=x(t)y(t)\\ y'(t)=x(t)y(t)\\ \end{cases}$$ To find the integral curves note that $y'(t)=x'(t)$ so there exists a constant $c_1$ such that $x(t)=y(t)+c_1$. So we get $y'-c_1y=y^2$ and this is a Bernoulli equation for wich I find the solution $y(t)=\dfrac{c_1}{c_1c_2e^{-c_1t}-1}$ and then we get $x(t)=\dfrac{c_1}{c_1c_2e^{-c_1t}-1}+c_1$. So the trajectory is given by curve $\gamma(t)=(\dfrac{c_1}{c_1c_2e^{-c_1t}-1}+c_1,\dfrac{c_1}{c_1c_2e^{-c_1t}-1})$ where $c_1$ and $c_2$ are any constant real numbers.
Now the integral curves are the support of the trajectories and are given as the lines $y=x-c_1$ where $c_1$ is any constant real number. Is my reasoning correct? and is there any easier way to do this?