Finding D.E. for part of an RC Circuit
Look at the first equation you have shown:
$$2*I_c + \frac{1}{4} \frac{dV_c}{dt} - y(t)=0$$
The first term is Voltage, the second term is Current, the third is Voltage. So the second term cannot be correct, it should simply be the voltage across the capacitor.
$$2*I_c + V_c - y(t)=0$$
There are three dependent variables and you only have two equations. So you need one more equation/relationship. Add the equation that relates current to voltage of a capacitor:
$$I_c =\frac{1}{4} \frac{dV_c}{dt}$$
Now you can take combine the three equations into one equation with a single dependent variable y(t) as requested by your problem statement.