How to analyze this diode circuit?
The circuit as shown is not viable - or you could analyze it in two phases, if you must: 

Phase 1:


Each 1n4148 diode is rated for 200 mA continuous, 450 mA peak repetitive current.
When wired as indicated, each diode will drop approximately 1 to 1.5 Volts (Fig.3 in datasheet) before the current exceeds absolute maximum rating
As the supply voltage is 5 Volts, this far exceeds the maximum 3 Volts noted above, so one of the two diodes will burn out.


Phase 2.a: If D2 burns out and becomes an open circuit:


There will be no voltage at Vout as D2 is now an open circuit
Result: Vout = 0 Volts


Phase 2.b: If D1 burns out and becomes an open circuit:


Vout = V1 - VD2 = ~ 4.4 Volts


Then there are the possibilities of D1 or D2 burning out to become short. That resultant analysis is left for you to do :-)