Capacitor, initial and boundary conditions, charging and discharging equation, 'EXO 8 Vidéo 2, 2 '

Two capacitors connected in series and supplied by an independent source of interval-variable current, Is(t). Knowing that the capacitors are initially uncharged, determine the expression of potentials V1(t) and V2(t) at nodes (1) and (2) of the circuit.
Method:
1) Current is interval variable, so we need to find the current for each time interval, i.e. :
Is1(t) for t=[0,1]
Is2(t) for t=[1;2].
Is(t) has a sawtooth shape made up of two straight-line segments, Is1(t) and Is2(t).
For t=[0;1] we obtain Is1(t)=0.06.t [A]
For t=[1;2] we get Is2(t)=-0.06.t+0.12 [A]
2) We have the fundamental relationship i(t)=C.dV(t)/dt , where V(t)=(1/C).{Integral of i(t).dt} C represents the capacitance of the combination of C1 and C2 in series, or C=2.10^-6[F].
3) For t=[0;1] we replace C by 2.10^-6 and i(t) by Is1(t), to calculate the corresponding interval voltage V1(t), i.e. V1(t)=(1/2.10^-6).{Integral of 0.06.t.dt}. Integration of V1(t) leads to V1(t)=15000.t²+K. The initial conditions for the voltage across the capacitors at t=0 are V1(0)=V2(0)=0, i.e. V1(0)=15000.(0)²+K=0, giving K=0.
4) The same reasoning for t=[1;2] leads to V2(t)= -15,000.t²+60,000.t+K, but this time we can no longer use the previous initial conditions of the voltage across the capacitors at t=0 to calculate K, as the variable t can no longer be Zero, t must belong to the interval t=[1;2] .
We therefore use the property of continuity of the voltage across the capacitor to find K: the value of the voltage across the capacitor cannot vary abruptly at time t=0.999[s] and at time t=1[s] , nor at time t=1.999[s] and at time t=2[s] , in other words, the continuity of the voltage translates into V1( 1[s] )=V2( 1[s] ), and it's this equality that enables us to calculate the value of K.
In fact, the current flowing through a capacitor can be interrupted abruptly, as when the circuit switch is opened at time t=100[s], for example. The capacitor is then no longer powered, so the current changes abruptly at time t=100[s], from i(100s)=certain current[A] to i(100s)=0[A], but the voltage across it remains unchanged just after the switch is opened at time t=100[s] , because the voltage across a capacitor at time t=100[s] results from the amount of charge accumulated over a period of time prior to time t=100[s] , and when we close the switch at time , say t=140[s] for example, the voltage across it is always the same, only just at time t=140 + 0.01[s] for example does the voltage across it change. For a capacitor, there is voltage continuity, but no current continuity. As for the graph representing the Is(t) current, we see a continuity of current at instants t=1[s] , t=2[s] and so on... there is no sudden break at instants t=1[s] , t=2[s] ... but this is the current supplied by the current source INDEPENDENT, the Is(t) current does not depend on the elements of the circuit, nor on the conditions in which the capacitor is located, nor on the opening or closing of a switch .
Note: for a coil of inductance L, we can exploit the property of continuity of the current flowing through the coil, but we can't exploit the continuity of the voltage across an inductance L, because that doesn't exist. The voltage across an inductance L varies abruptly, and its polarity may be reversed. The reaction of a coil to variations in current and voltage is exactly the opposite of that of a capacitor. This is why the combination of the two - a capacitor C and an inductor L - makes it possible to COMPENSATE reactive power or power factor.
C' compensates for what 'L' can't have, and 'L' compensates for what 'C' can't have.