Diode circuit analysis solution
Guess which diodes are conducting. Remove the non-conducting diodes, and replace the conducting diodes with a 0.7V voltage source, + side at the anode. Solve using standard analysis.

Then, verify that your guess was correct. Current should flow into the + side of the simulated conducting diode, if not, it wasn't conducting. Where the non-conducting diode was should have a voltage less than 0.7V (anode V - cathode V), if not it should not have been removed.

For this exercise, it is fairly obvious that current is flowing from top to bottom, so D2 is probably conducting. Replace D2 with a voltage source, then solve. Verify that current is flowing into the anode of D2. Then verify that V4 - V1 is not greater than 0.7V.

If your guesses were wrong (verification didn't work), then make another guess and solve again.