How analyze this circuit?
This is a model of full wave rectifier with high source resistance(1kOhm). It has 1uF filtering capacitor and a 50kOhm load. As commented, the charging of the cap is quite fast, only 1ms time constant. The discharging isn't fast due 50ms time constant. You can well assume that the cap charges to the available peak voltage (=AC amplitude - 0,7 volts) AC input dives, but the cap discharges slowly. It starts to charge again when the next rising sine(minus 0,7 volts) meets the decaying voltage of the cap.

Finding where the cap starts to get charged again needs a transcendental equation, which can be solved only iteratively or graphically.  Draw the waveforms and see where the curves meet. Express this as the principle and find more exact numbers with a circuit simulator. It can take the diode into the account more accurately than only assuming 0,7V drop.

The starting transient is different. You must assume initially 0V in the cap. The simulation shows that 4,3V isnt reached in the first half cycle. The simulation also shows that 0,7V drop in the diode is underestimated. There's also some drop in 1kOhm resistors due the 50kOhm load - it's a voltage divider which drops about 2% off.