Solving following System of ordinary differential equations.

Question

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) ?

Answer 1

Hint...you can subtract the equations and write $u=y-x$ to obtain a seperable variable DE in $u$ and $t$

Answer 2

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 $y(t)$

$$y' +y(1+t)= K_1te^{\frac {t^2}2-t}$$