Take V
t as 25.85 mV.
Plug your specified Vf and the current If it was specified at into the simplified Shockley equation
If=Ise(Vf/Vt)Solve for Is for both diodes.
Then estimate the possible current through the diodes. The one on the right looks like it dominates so solve that first. 12V/(415R+100R)=23.3mA as a first estimate. Plug that back into the diode equation and you get 0.7V (0.703) for Vf if it got all the current.
That would give you 2.33 + 0.7V at the juntion if there was no current through the other diode.
Now assume the other diode is a dead short - the 500R and the 415R form a potential divider with a Thevenin equivalent voltage of 6.56V and equivalent resistance of 227R
Recalculate: 6.56V/(237R+100R)=20.1mA, and plug that back into the diode equation for the right diode to get 0.699V for Vf - a negligible change.
Now treat the 415R and 100R resistors as a potential divider between two sources, 12V and 0.7V and find the Thevenin equivalent voltage of 2.89V and equivalent resistance of 80.6R. The max current through the left diode cant be more than 2.89V/(80.6R+500R)= 5.0mA. Plug that into the diode equation, and you get a Vf of 2.06V. Re-solve for If: (2.89V-2.06V)/(80.6R+500R)=1.43mA, and back into the diode equation for a revised Vf of 2.03V, again a negligable change.
At this point, you have a good estimate for If and Vf of the left diode, but If for the right diode is still wrong, although you know its Vf must be 0.7V
Now you have Vf for both diodes at close to their actual If, you can treat them as pure voltage sources and finish solving the circuit by superposition.
Then throw the result out of the window if you need better than a ballpark estimate as you *ASS*U*ME*ed the diodes were ideal with no evidence to support that assumption, and also have made absolutely no allowance for the dissipation in each diode increasing the junction temperature and decreasing Vf.
Due to the need to repeatedly solve the diode equation it goes a lot easier if you use a spreadsheet rather than doing it the old-skool way with pencil and paper and a
slipstick.