one could argue that the voltage source has the same behaviour as a capacity C1 of earlier lesson where 2 capacitors were involved. The voltage source has the behaviour of a Capacity C1 that cannot be decharged, or otherwise said is very very large this results into Vout (at t = infinite) = V1* C1 / C1 + C2 with very big C1 this becomes Vout (at t = infinite) = V1
to filter out solution what we can get out of equations we solved for t < 0. We start our analysis in t=0 with given initial conditions. And even if our solved equations can be plot for t