Diode circuit with varying voltage
Using the simplified diode model (fixed Vf of 0.7 volts, say), you can see there are two possible situations- diode conducting and diode not conducting. When the diode is not conducting you replace it with an open circuit and analyze easily. 

When the diode is conducting, replace it with a voltage source of Vf, and the analysis is not much more difficult. 

It should be easy to see that the diode just starts to conduct when the forward voltage is exactly equal to Vf without it conducting - so the first analysis will give you that point. Since there is no current, the voltage across R2 is 0, and the equation is just Vf= Vi*6/7. 

So you have:

case 1: Vo = f(Vi)  valid for Vi &lt;= 7/6 * Vf

case 2: Vo = f'(Vi)  valid for Vi >= 7/6*Vf 

If your input voltage is always > 1V you may only have to deal with one situation.