Solving following System of ordinary differential equations. How to solve system of ODEs which contains independent variable like t in equation as in this particular case : 
$$\frac{dx}{dt}= -x + ty $$ $$ \frac{dy}{dt} = tx-y$$ can we solve them or additional information may be needed (may be about t) ?

Sum both equations to get
$$x'+y'=t(x+y)-(x+y)$$
Solve for $z=x+y$ it's seperable
$$z'=z(t-1)$$
$$\ln|z|=\frac {t^2}2-t+K_1$$
$$x=K_1e^{\frac {t^2}2-t}-y$$
Then solve one of your two equations..with $x(t)$ to get $y(t)$
$$y' +y(1+t)= K_1te^{\frac {t^2}2-t}$$
